Complete Matrix Shapes, Transposes, and Index Guards
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261012-advance-20261016-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Finite matrices: dimensions, indexing, and transpose · qf-density16-matrix-candidate-transpose-audit · candidate
Canonical question JSONL · English JSONL · Validation report