# Signed Real 2×2 Determinants and Product Orders

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r04-determinant-0100/)

Determinants of real 2×2 matrices · qf-density16-det-candidate-absolute-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r04-determinant-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r04-determinant-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r04-determinant-foundations/validation-report.json)