Exact RREF Rank, Minor Bounds and Singular-Determinant Limits
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batch-20261014-advance-20261103-r01-matrix-rank-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
2. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 1], [0, 2]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
3. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 2], [0, 3]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
4. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 3], [0, 4]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
5. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 4], [0, 5]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
6. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 5], [0, 6]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
7. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 6], [0, 7]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
8. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 7], [0, 8]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
9. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 8], [0, 9]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
10. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 9], [0, 10]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
11. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 10], [0, 11]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
12. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 11], [0, 12]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
13. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 12], [0, 13]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
14. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 13], [0, 14]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
15. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 14], [0, 15]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
16. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 15], [0, 16]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
17. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 16], [0, 17]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
18. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 17], [0, 18]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
19. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 18], [0, 19]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
20. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 19], [0, 20]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
21. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 20], [0, 21]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
22. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 21], [0, 22]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
23. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 22], [0, 23]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
24. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 23], [0, 24]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
25. The Complete Exact RREF Supplies Matrix Rank
Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (matrix,rref,pivot_columns,rank,rank_preserving_exact_row_operations), with zero-based pivot columns in increasing order.
View answer and explanation
([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations)
Complete exact elimination through every available pivot. The number of pivot columns is the rank; retain all rows, including zero rows. Do not stop after a few successful row operations and treat them as a full upper bound. The exact result, retaining the requested order, is ([[1, 24], [0, 25]], [[1, 0], [0, 1]], [0, 1], 2, rank_preserving_exact_row_operations).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-exact-rref · candidate
26. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
27. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
28. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
29. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
30. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
31. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
32. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
33. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
34. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
35. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
36. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
37. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
38. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
39. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
40. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
41. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
42. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
43. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
44. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
45. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
46. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
47. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
48. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
49. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
50. A Nonzero Minor Supplies a Lower Bound, Not Automatically Exact Rank
Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Selected distinct row indices=[0] and equally many distinct column indices=[0] define a nonempty square minor in those supplied orders. Give (rows,cols,minor,determinant,nonzero?,lower_bound).
View answer and explanation
([0], [0], [[1]], 1, Yes, rank_at_least_1)
Form the exact minor and its determinant. A nonzero r-by-r minor yields rank_at_least_r with the numeric size; a zero minor yields no_lower_bound_from_zero_minor, not a conclusion that the whole matrix has rank zero. The exact result, retaining the requested order, is ([0], [0], [[1]], 1, Yes, rank_at_least_1).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-minor-lower-bound · candidate
51. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 0], [0, 1]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 1, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 1, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
52. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 1], [0, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 2, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 2, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
53. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 2], [0, 3]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 3, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 3, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
54. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 3], [0, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 4, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 4, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
55. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 4], [0, 5]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 5, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 5, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
56. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 5], [0, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 6, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 6, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
57. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 6], [0, 7]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 7, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 7, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
58. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 7], [0, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 8, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 8, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
59. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 8], [0, 9]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 9, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 9, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
60. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 9], [0, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 10, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 10, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
61. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 10], [0, 11]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 11, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 11, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
62. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 11], [0, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 12, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 12, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
63. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 12], [0, 13]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 13, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 13, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
64. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 13], [0, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 14, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 14, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
65. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 14], [0, 15]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 15, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 15, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
66. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 15], [0, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 16, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 16, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
67. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 16], [0, 17]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 17, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 17, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
68. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 17], [0, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 18, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 18, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
69. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 18], [0, 19]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 19, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 19, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
70. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 19], [0, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 20, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 20, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
71. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 20], [0, 21]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 21, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 21, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
72. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 21], [0, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 22, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 22, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
73. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 22], [0, 23]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 23, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 23, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
74. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 23], [0, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 24, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 24, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
75. Full Rank Needs the Correct Square-Matrix Certificate
Use the complete ordered integer matrix A=[[1, 24], [0, 25]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (size,determinant,nonzero?,rank,full_rank?).
View answer and explanation
(2, 25, Yes, 2, Yes)
Compute the determinant and full exact RREF rank separately. For this square matrix, full rank agrees with nonzero determinant; a zero determinant still leaves several possible lower ranks. The exact result, retaining the requested order, is (2, 25, Yes, 2, Yes).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-full-rank-audit · candidate
76. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 1], [2, 2]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
77. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 2], [2, 4]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
78. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 3], [2, 6]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
79. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 4], [2, 8]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
80. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 5], [2, 10]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
81. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 6], [2, 12]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
82. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 7], [2, 14]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
83. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 8], [2, 16]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
84. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 9], [2, 18]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
85. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 10], [2, 20]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
86. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 11], [2, 22]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
87. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 12], [2, 24]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
88. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 13], [2, 26]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
89. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 14], [2, 28]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
90. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 15], [2, 30]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
91. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 16], [2, 32]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
92. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 17], [2, 34]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
93. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 18], [2, 36]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
94. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 19], [2, 38]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
95. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 20], [2, 40]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
96. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 21], [2, 42]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
97. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 22], [2, 44]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
98. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 23], [2, 46]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
99. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 24], [2, 48]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
100. Zero Determinant Does Not Determine Which Lower Rank Occurs
Use the complete ordered integer matrix A=[[1, 25], [2, 50]], limited to square 2-by-2 or 3-by-3 matrices. RREF uses exact rational, rank-preserving row operations. A nonzero square minor gives only a lower bound; an exact rank needs a matching upper-bound certificate such as the full RREF. A zero determinant excludes full rank but does not by itself fix the rank. Equality of row rank and column rank invokes the theorem, not sampled operations. Give (determinant,zero?,rank,consequence,zero_determinant_does_not_determine_exact_rank). consequence is det_zero_only_excludes_full_rank or premise_false.
View answer and explanation
(0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank)
Retain the exact rank obtained from RREF alongside the determinant flag. The zero matrix and a nonzero singular matrix may both have determinant zero but different ranks. A nonzero determinant marks the zero-determinant premise false, not a failed theorem. The exact result, retaining the requested order, is (0, Yes, 1, det_zero_only_excludes_full_rank, zero_determinant_does_not_determine_exact_rank).
Reviewed step-by-step derivation
Matrix rank · qf-mod34-rank-determinant-zero-boundary · candidate
Canonical question JSONL · English JSONL · Validation report