Signed Real 2×2 Determinants and Product Orders
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261012-advance-20261016-r04-determinant-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], 0, [[0, 0], [0, 0]], 0, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
2. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, [[3, 4], [1, 2]], 2, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
3. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-10, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-10, 1], [2, 1]], -12, [[2, 1], [-10, 1]], 12, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
4. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-9, 1], [2, 1]], -11, [[2, 1], [-9, 1]], 11, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
5. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 1], [2, 1]], -10, [[2, 1], [-8, 1]], 10, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
6. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], -9, [[2, 1], [-7, 1]], 9, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
7. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], -8, [[2, 1], [-6, 1]], 8, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
8. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], -7, [[2, 1], [-5, 1]], 7, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
9. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], -6, [[2, 1], [-4, 1]], 6, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
10. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], -5, [[2, 1], [-3, 1]], 5, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
11. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], -4, [[2, 1], [-2, 1]], 4, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
12. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], -3, [[2, 1], [-1, 1]], 3, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
13. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], -2, [[2, 1], [0, 1]], 2, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
14. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], -1, [[2, 1], [1, 1]], 1, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
15. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], 0, [[2, 1], [2, 1]], 0, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
16. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], 1, [[2, 1], [3, 1]], -1, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
17. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], 2, [[2, 1], [4, 1]], -2, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
18. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], 3, [[2, 1], [5, 1]], -3, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
19. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], 4, [[2, 1], [6, 1]], -4, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
20. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], 5, [[2, 1], [7, 1]], -5, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
21. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], 6, [[2, 1], [8, 1]], -6, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
22. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], 7, [[2, 1], [9, 1]], -7, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
23. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], 8, [[2, 1], [10, 1]], -8, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
24. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], 9, [[2, 1], [11, 1]], -9, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
25. Signed Determinants Before and After a Row Swap
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]]. Give (M, D(M), matrix with its two rows swapped, swapped determinant, sum of the two determinants).
View answer and explanation
([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0)
Keep row/column positions and signs, swap the actual rows, and evaluate the 2×2 product difference. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], 10, [[2, 1], [12, 1]], -10, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-row-swap-signed · candidate
26. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], t=0. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(0, [[0, 0], [0, 0]], 0, 0)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [[0, 0], [0, 0]], 0, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
27. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], t=0. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-2, [[0, 0], [0, 0]], 0, 0)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2, [[0, 0], [0, 0]], 0, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
28. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(1, [[-2, 0], [0, -2]], 4, 4)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[-2, 0], [0, -2]], 4, 4).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
29. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], t=9007199254740991. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[9007199254740991, 0], [0, 9007199254740991]], 81129638414606663681390495662081, 81129638414606663681390495662081).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
30. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-10, [[-8, 1], [2, 1]], -10, -10)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-10, [[-8, 1], [2, 1]], -10, -10).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
31. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-9, [[14, -2], [-4, -2]], -36, -36)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-9, [[14, -2], [-4, -2]], -36, -36).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
32. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-8, [[-6, 1], [2, 1]], -8, -8)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-8, [[-6, 1], [2, 1]], -8, -8).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
33. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-7, [[10, -2], [-4, -2]], -28, -28)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-7, [[10, -2], [-4, -2]], -28, -28).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
34. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-6, [[-4, 1], [2, 1]], -6, -6)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-6, [[-4, 1], [2, 1]], -6, -6).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
35. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-5, [[6, -2], [-4, -2]], -20, -20)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-5, [[6, -2], [-4, -2]], -20, -20).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
36. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-4, [[-2, 1], [2, 1]], -4, -4)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-4, [[-2, 1], [2, 1]], -4, -4).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
37. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-3, [[2, -2], [-4, -2]], -12, -12)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3, [[2, -2], [-4, -2]], -12, -12).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
38. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-2, [[0, 1], [2, 1]], -2, -2)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2, [[0, 1], [2, 1]], -2, -2).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
39. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(-1, [[-2, -2], [-4, -2]], -4, -4)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1, [[-2, -2], [-4, -2]], -4, -4).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
40. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(0, [[2, 1], [2, 1]], 0, 0)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [[2, 1], [2, 1]], 0, 0).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
41. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(1, [[-6, -2], [-4, -2]], 4, 4)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [[-6, -2], [-4, -2]], 4, 4).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
42. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(2, [[4, 1], [2, 1]], 2, 2)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[4, 1], [2, 1]], 2, 2).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
43. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(3, [[-10, -2], [-4, -2]], 12, 12)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[-10, -2], [-4, -2]], 12, 12).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
44. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(4, [[6, 1], [2, 1]], 4, 4)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[6, 1], [2, 1]], 4, 4).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
45. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(5, [[-14, -2], [-4, -2]], 20, 20)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[-14, -2], [-4, -2]], 20, 20).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
46. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(6, [[8, 1], [2, 1]], 6, 6)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[8, 1], [2, 1]], 6, 6).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
47. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(7, [[-18, -2], [-4, -2]], 28, 28)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (7, [[-18, -2], [-4, -2]], 28, 28).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
48. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(8, [[10, 1], [2, 1]], 8, 8)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (8, [[10, 1], [2, 1]], 8, 8).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
49. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]], t=-2. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(9, [[-22, -2], [-4, -2]], 36, 36)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (9, [[-22, -2], [-4, -2]], 36, 36).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
50. Complete Matrix Scaling and Signed Determinant Routes
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]], t=1. Give (D(M), tM, D(tM), t²D(M)); use the specified value even for negative or zero t.
View answer and explanation
(10, [[12, 1], [2, 1]], 10, 10)
Scale every matrix entry, then calculate its determinant; independently multiply the original signed determinant by t². These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (10, [[12, 1], [2, 1]], 10, 10).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-scaled-matrix · candidate
51. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 0], [0, 1]], N=[[1, 0], [0, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, Yes)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [[1, 0], [0, 1]], 1, 1, 1, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
52. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [0, 1]], N=[[1, 0], [1, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [1, 1]], [[1, 1], [1, 2]], 1, 1, 1, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
53. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], N=[[1, 2], [3, 4]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, Yes)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], [[0, 0], [0, 0]], 0, 0, 0, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
54. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], [[-9, 1], [-18, 3]], -9, -9, -9, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
55. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], [[-8, 1], [-16, 3]], -8, -8, -8, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
56. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], [[-7, 1], [-14, 3]], -7, -7, -7, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
57. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], [[-6, 1], [-12, 3]], -6, -6, -6, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
58. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], [[-5, 1], [-10, 3]], -5, -5, -5, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
59. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], [[-4, 1], [-8, 3]], -4, -4, -4, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
60. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], [[-3, 1], [-6, 3]], -3, -3, -3, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
61. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], [[-2, 1], [-4, 3]], -2, -2, -2, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
62. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], [[-1, 1], [-2, 3]], -1, -1, -1, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
63. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], [[0, 1], [0, 3]], 0, 0, 0, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
64. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], [[1, 1], [2, 3]], 1, 1, 1, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
65. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], [[2, 1], [4, 3]], 2, 2, 2, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
66. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
67. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], [[4, 1], [8, 3]], 4, 4, 4, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
68. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], [[5, 1], [10, 3]], 5, 5, 5, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
69. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], [[6, 1], [12, 3]], 6, 6, 6, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
70. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], [[7, 1], [14, 3]], 7, 7, 7, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
71. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], [[8, 1], [16, 3]], 8, 8, 8, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
72. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], [[9, 1], [18, 3]], 9, 9, 9, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
73. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], [[10, 1], [20, 3]], 10, 10, 10, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
74. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[13, 1], [2, 1]], [[11, 1], [22, 3]], 11, 11, 11, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
75. Both Matrix Product Orders, Determinants, and Equality
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).
View answer and explanation
([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, No)
Compute each ordinary product order separately and compare full entries; matching determinant values do not establish equal matrices or general commutativity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[14, 1], [2, 1]], [[12, 1], [24, 3]], 12, 12, 12, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-product-two-orders · candidate
76. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 0], [0, 0]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[0, 0], [0, 0]], 0, 0, 0, Yes, No)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 0]], 0, 0, 0, Yes, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
77. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[1, 2], [3, 4]], -2, 2, 4, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, 2, 4, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
78. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 2], [3, 4]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[1, 2], [3, 4]], -2, 2, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [3, 4]], -2, 2, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
79. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-9, 1], [2, 1]], integer candidate=-10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-9, 1], [2, 1]], -11, 11, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-9, 1], [2, 1]], -11, 11, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
80. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-8, 1], [2, 1]], integer candidate=-10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-8, 1], [2, 1]], -10, 10, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 1], [2, 1]], -10, 10, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
81. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-7, 1], [2, 1]], integer candidate=-8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-7, 1], [2, 1]], -9, 9, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 1], [2, 1]], -9, 9, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
82. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-6, 1], [2, 1]], integer candidate=-8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-6, 1], [2, 1]], -8, 8, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 1], [2, 1]], -8, 8, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
83. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-5, 1], [2, 1]], integer candidate=-6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-5, 1], [2, 1]], -7, 7, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 1], [2, 1]], -7, 7, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
84. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-4, 1], [2, 1]], integer candidate=-6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-4, 1], [2, 1]], -6, 6, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 1], [2, 1]], -6, 6, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
85. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-3, 1], [2, 1]], integer candidate=-4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-3, 1], [2, 1]], -5, 5, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 1], [2, 1]], -5, 5, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
86. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-2, 1], [2, 1]], integer candidate=-4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-2, 1], [2, 1]], -4, 4, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 1], [2, 1]], -4, 4, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
87. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[-1, 1], [2, 1]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[-1, 1], [2, 1]], -3, 3, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 1], [2, 1]], -3, 3, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
88. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[0, 1], [2, 1]], integer candidate=-2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[0, 1], [2, 1]], -2, 2, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [2, 1]], -2, 2, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
89. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[1, 1], [2, 1]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[1, 1], [2, 1]], -1, 1, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1], [2, 1]], -1, 1, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
90. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[2, 1], [2, 1]], integer candidate=0. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[2, 1], [2, 1]], 0, 0, 0, Yes, No)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 1], [2, 1]], 0, 0, 0, Yes, No).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
91. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[3, 1], [2, 1]], 1, 1, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 1], [2, 1]], 1, 1, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
92. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[4, 1], [2, 1]], integer candidate=2. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 1], [2, 1]], 2, 2, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
93. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[5, 1], [2, 1]], integer candidate=4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[5, 1], [2, 1]], 3, 3, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 1], [2, 1]], 3, 3, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
94. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[6, 1], [2, 1]], integer candidate=4. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[6, 1], [2, 1]], 4, 4, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 1], [2, 1]], 4, 4, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
95. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[7, 1], [2, 1]], integer candidate=6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[7, 1], [2, 1]], 5, 5, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 1], [2, 1]], 5, 5, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
96. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[8, 1], [2, 1]], integer candidate=6. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[8, 1], [2, 1]], 6, 6, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 1], [2, 1]], 6, 6, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
97. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[9, 1], [2, 1]], integer candidate=8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[9, 1], [2, 1]], 7, 7, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 1], [2, 1]], 7, 7, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
98. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[10, 1], [2, 1]], integer candidate=8. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[10, 1], [2, 1]], 8, 8, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 1], [2, 1]], 8, 8, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
99. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[11, 1], [2, 1]], integer candidate=10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[11, 1], [2, 1]], 9, 9, 1, No, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 1], [2, 1]], 9, 9, 1, No, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
100. Signed Determinant versus Absolute-Value Candidate
Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[12, 1], [2, 1]], integer candidate=10. Give (M, signed D, abs(D), candidate−signed D, candidate correctness, invertibility over R).
View answer and explanation
([[12, 1], [2, 1]], 10, 10, 0, Yes, Yes)
Retain the signed product difference for candidate checking and independently require D≠0 for real invertibility; an absolute value cannot replace a negative determinant. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 1], [2, 1]], 10, 10, 0, Yes, Yes).
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate
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