# Complete Matrix Shapes, Transposes, and Index Guards

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-r02-matrix-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. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([1, 1], [[0]], [[0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [[0]], [[0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0001/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 2. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([1, 2], [[-11], [-12]], [[-11, -12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [[-11], [-12]], [[-11, -12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0002/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 3. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0003/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 4. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-9, -10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0004/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 5. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, -9, -10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0005/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 6. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7], [-6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([2, 1], [[-7, -6]], [[-7], [-6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [[-7, -6]], [[-7], [-6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0006/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 7. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, -7], [-5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0007/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 8. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, -6, -7], [-4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0008/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 9. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0009/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 10. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0010/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 11. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2], [-1], [0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0011/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 12. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, -2], [0, -1], [1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0012/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 13. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0013/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 14. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0014/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 15. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0015/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 16. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0016/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 17. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 3], [5, 4], [6, 5], [7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0017/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 18. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0018/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 19. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0019/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 20. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0020/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 21. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8], [9], [10], [11], [12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0021/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 22. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0022/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 23. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0023/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 24. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0024/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 25. Matrix Shape and Complete Transpose Round Trip

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).

View answer and explanation([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], Yes)

Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0025/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-transpose-roundtrip · candidate





## 26. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[0]], N=[[0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([1, 1], [[0]], [[0]], [[0]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [[0]], [[0]], [[0]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0026/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 27. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-11, -12]], N=[[0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0027/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 28. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-10, -11, -12]], N=[[0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0028/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 29. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-9, -10, -11, -12]], N=[[0, 1, 2, 3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0029/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 30. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-8, -9, -10, -11, -12]], N=[[0, 1, 2, 3, 4]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0030/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 31. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-7], [-6]], N=[[0], [-1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0031/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 32. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-6, -7], [-5, -6]], N=[[0, 1], [-1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0032/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 33. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-5, -6, -7], [-4, -5, -6]], N=[[0, 1, 2], [-1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0033/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 34. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0034/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 35. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0035/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 36. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-2], [-1], [0]], N=[[0], [-1], [-2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0036/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 37. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-1, -2], [0, -1], [1, 0]], N=[[0, 1], [-1, 0], [-2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0037/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 38. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 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-r02-matrix-0038/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 39. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0039/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 40. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0040/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 41. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[3], [4], [5], [6]], N=[[0], [-1], [-2], [-3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0041/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 42. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[4, 3], [5, 4], [6, 5], [7, 6]], N=[[0, 1], [-1, 0], [-2, -1], [-3, -2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0042/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 43. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0043/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 44. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0044/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 45. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0045/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 46. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[8], [9], [10], [11], [12]], N=[[0], [-1], [-2], [-3], [-4]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0046/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 47. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], N=[[0, 1], [-1, 0], [-2, -1], [-3, -2], [-4, -3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0047/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 48. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1], [-4, -3, -2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0048/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 49. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0], [-4, -3, -2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0049/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 50. Same-Shape Matrix Sum and Both Transpose Routes

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1], [-4, -3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

View answer and explanation([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])

Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0050/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-sum-transpose · candidate





## 51. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], ordered (i,j) candidates=[[1, 1], [1, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([1, 1], [Yes, Yes, No], [0, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [Yes, Yes, No], [0, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0051/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 52. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-11, -12]], ordered (i,j) candidates=[[1, 1], [1, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([1, 2], [Yes, Yes, No], [-11, -12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [Yes, Yes, No], [-11, -12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0052/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 53. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([1, 3], [Yes, Yes, No], [-10, -12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [Yes, Yes, No], [-10, -12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0053/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 54. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-9, -10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([1, 4], [Yes, Yes, No], [-9, -12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [Yes, Yes, No], [-9, -12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0054/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 55. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, -9, -10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([1, 5], [Yes, Yes, No], [-8, -12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [Yes, Yes, No], [-8, -12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0055/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 56. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7], [-6]], ordered (i,j) candidates=[[1, 1], [2, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([2, 1], [Yes, Yes, No], [-7, -6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [Yes, Yes, No], [-7, -6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0056/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 57. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, -7], [-5, -6]], ordered (i,j) candidates=[[1, 1], [2, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([2, 2], [Yes, Yes, No], [-6, -6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [Yes, Yes, No], [-6, -6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0057/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 58. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, -6, -7], [-4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([2, 3], [Yes, Yes, No], [-5, -6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [Yes, Yes, No], [-5, -6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0058/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 59. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([2, 4], [Yes, Yes, No], [-4, -6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [Yes, Yes, No], [-4, -6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0059/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 60. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([2, 5], [Yes, Yes, No], [-3, -6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [Yes, Yes, No], [-3, -6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0060/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 61. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2], [-1], [0]], ordered (i,j) candidates=[[1, 1], [3, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([3, 1], [Yes, Yes, No], [-2, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [Yes, Yes, No], [-2, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0061/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 62. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, -2], [0, -1], [1, 0]], ordered (i,j) candidates=[[1, 1], [3, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([3, 2], [Yes, Yes, No], [-1, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, No], [-1, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0062/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 63. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([3, 3], [Yes, Yes, No], [0, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [Yes, Yes, No], [0, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0063/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 64. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([3, 4], [Yes, Yes, No], [1, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [Yes, Yes, No], [1, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0064/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 65. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([3, 5], [Yes, Yes, No], [2, 0, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [Yes, Yes, No], [2, 0, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0065/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 66. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]], ordered (i,j) candidates=[[1, 1], [4, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([4, 1], [Yes, Yes, No], [3, 6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [Yes, Yes, No], [3, 6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0066/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 67. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 3], [5, 4], [6, 5], [7, 6]], ordered (i,j) candidates=[[1, 1], [4, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([4, 2], [Yes, Yes, No], [4, 6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [Yes, Yes, No], [4, 6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0067/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 68. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([4, 3], [Yes, Yes, No], [5, 6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [Yes, Yes, No], [5, 6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0068/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 69. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([4, 4], [Yes, Yes, No], [6, 6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [Yes, Yes, No], [6, 6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0069/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 70. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([4, 5], [Yes, Yes, No], [7, 6, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [Yes, Yes, No], [7, 6, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0070/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 71. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8], [9], [10], [11], [12]], ordered (i,j) candidates=[[1, 1], [5, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([5, 1], [Yes, Yes, No], [8, 12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [Yes, Yes, No], [8, 12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0071/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 72. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], ordered (i,j) candidates=[[1, 1], [5, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([5, 2], [Yes, Yes, No], [9, 12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [Yes, Yes, No], [9, 12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0072/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 73. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([5, 3], [Yes, Yes, No], [10, 12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [Yes, Yes, No], [10, 12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0073/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 74. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([5, 4], [Yes, Yes, No], [11, 12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [Yes, Yes, No], [11, 12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0074/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 75. Three 1-Based Matrix Queries with Eligibility Guards

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.

View answer and explanation([5, 5], [Yes, Yes, No], [12, 12, invalid-index])

Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [Yes, Yes, No], [12, 12, invalid-index]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0075/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-three-index-qualification · candidate





## 76. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], candidate=[[0]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[0]], [1, 1], Yes, [[0]], Yes)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0]], [1, 1], Yes, [[0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0076/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 77. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2]], candidate=[[1, 2]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[1], [2]], [1, 2], No, shape-mismatch, No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1], [2]], [1, 2], No, shape-mismatch, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0077/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 78. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2], [3, 4]], candidate=[[1, 3], [2, 4]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[1, 3], [2, 4]], [2, 2], Yes, [[0, 0], [0, 0]], Yes)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 3], [2, 4]], [2, 2], Yes, [[0, 0], [0, 0]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0078/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 79. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2], [3, 4]], candidate=[[1, 0], [2, 4]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[1, 3], [2, 4]], [2, 2], Yes, [[0, -3], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 3], [2, 4]], [2, 2], Yes, [[0, -3], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0079/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 80. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, 1], [2, -1]], candidate=[[-7, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0080/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 81. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7, 1], [2, -1]], candidate=[[-6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0081/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 82. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, 1], [2, -1]], candidate=[[-5, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0082/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 83. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, 1], [2, -1]], candidate=[[-4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0083/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 84. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, 1], [2, -1]], candidate=[[-3, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0084/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 85. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, 1], [2, -1]], candidate=[[-2, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0085/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 86. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2, 1], [2, -1]], candidate=[[-1, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0086/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 87. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, 1], [2, -1]], candidate=[[0, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[-1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0087/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 88. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, 1], [2, -1]], candidate=[[1, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[0, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0088/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 89. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 1], [2, -1]], candidate=[[2, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0089/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 90. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1], [2, -1]], candidate=[[3, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0090/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 91. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3, 1], [2, -1]], candidate=[[4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0091/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 92. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 1], [2, -1]], candidate=[[5, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0092/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 93. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 1], [2, -1]], candidate=[[6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0093/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 94. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 1], [2, -1]], candidate=[[7, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0094/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 95. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 1], [2, -1]], candidate=[[8, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0095/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 96. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8, 1], [2, -1]], candidate=[[9, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0096/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 97. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 1], [2, -1]], candidate=[[10, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[9, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0097/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 98. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 1], [2, -1]], candidate=[[11, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[10, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0098/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 99. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 1], [2, -1]], candidate=[[12, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[11, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0099/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate





## 100. Candidate Transpose Shape before Entrywise Residuals

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 1], [2, -1]], candidate=[[13, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).

View answer and explanation([[12, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)

Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r02-matrix-0100/)

Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r02-matrix-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r02-matrix-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r02-matrix-foundations/validation-report.json)