Question 7640: Same-Shape Matrix Sum and Both Transpose Routes

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-matrix

Step-by-step derivation

M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]];shape=[3, 5]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=2

target(1,1)=2

M transpose:source(1,2)=1

target(2,1)=1

M transpose:source(1,3)=0

target(3,1)=0

M transpose:source(1,4)=-1

target(4,1)=-1

M transpose:source(1,5)=-2

target(5,1)=-2

M transpose:source(2,1)=3

target(1,2)=3

M transpose:source(2,2)=2

target(2,2)=2

M transpose:source(2,3)=1

target(3,2)=1

M transpose:source(2,4)=0

target(4,2)=0

M transpose:source(2,5)=-1

target(5,2)=-1

M transpose:source(3,1)=4

target(1,3)=4

M transpose:source(3,2)=3

target(2,3)=3

M transpose:source(3,3)=2

target(3,3)=2

M transpose:source(3,4)=1

target(4,3)=1

M transpose:source(3,5)=0

target(5,3)=0

M shape=[3, 5];N shape=[3, 5]

same shape;entrywise sum permitted

M+N(1,1) := sum(2, 0)

M+N(1,1) = 2

M+N(1,2) := sum(1, 1)

M+N(1,2) = 2

M+N(1,3) := sum(0, 2)

M+N(1,3) = 2

M+N(1,4) := sum(-1, 3)

M+N(1,4) = 2

M+N(1,5) := sum(-2, 4)

M+N(1,5) = 2

M+N(2,1) := sum(3, -1)

M+N(2,1) = 2

M+N(2,2) := sum(2, 0)

M+N(2,2) = 2

M+N(2,3) := sum(1, 1)

M+N(2,3) = 2

M+N(2,4) := sum(0, 2)

M+N(2,4) = 2

M+N(2,5) := sum(-1, 3)

M+N(2,5) = 2

M+N(3,1) := sum(4, -2)

M+N(3,1) = 2

M+N(3,2) := sum(3, -1)

M+N(3,2) = 2

M+N(3,3) := sum(2, 0)

M+N(3,3) = 2

M+N(3,4) := sum(1, 1)

M+N(3,4) = 2

M+N(3,5) := sum(0, 2)

M+N(3,5) = 2

sum transpose:source(1,1)=2

target(1,1)=2

sum transpose:source(1,2)=2

target(2,1)=2

sum transpose:source(1,3)=2

target(3,1)=2

sum transpose:source(1,4)=2

target(4,1)=2

sum transpose:source(1,5)=2

target(5,1)=2

sum transpose:source(2,1)=2

target(1,2)=2

sum transpose:source(2,2)=2

target(2,2)=2

sum transpose:source(2,3)=2

target(3,2)=2

sum transpose:source(2,4)=2

target(4,2)=2

sum transpose:source(2,5)=2

target(5,2)=2

sum transpose:source(3,1)=2

target(1,3)=2

sum transpose:source(3,2)=2

target(2,3)=2

sum transpose:source(3,3)=2

target(3,3)=2

sum transpose:source(3,4)=2

target(4,3)=2

sum transpose:source(3,5)=2

target(5,3)=2

N transpose:source(1,1)=0

target(1,1)=0

N transpose:source(1,2)=1

target(2,1)=1

N transpose:source(1,3)=2

target(3,1)=2

N transpose:source(1,4)=3

target(4,1)=3

N transpose:source(1,5)=4

target(5,1)=4

N transpose:source(2,1)=-1

target(1,2)=-1

N transpose:source(2,2)=0

target(2,2)=0

N transpose:source(2,3)=1

target(3,2)=1

N transpose:source(2,4)=2

target(4,2)=2

N transpose:source(2,5)=3

target(5,2)=3

N transpose:source(3,1)=-2

target(1,3)=-2

N transpose:source(3,2)=-1

target(2,3)=-1

N transpose:source(3,3)=0

target(3,3)=0

N transpose:source(3,4)=1

target(4,3)=1

N transpose:source(3,5)=2

target(5,3)=2

MT+NT(1,1) := sum(2, 0)

MT+NT(1,1) = 2

MT+NT(1,2) := sum(3, -1)

MT+NT(1,2) = 2

MT+NT(1,3) := sum(4, -2)

MT+NT(1,3) = 2

MT+NT(2,1) := sum(1, 1)

MT+NT(2,1) = 2

MT+NT(2,2) := sum(2, 0)

MT+NT(2,2) = 2

MT+NT(2,3) := sum(3, -1)

MT+NT(2,3) = 2

MT+NT(3,1) := sum(0, 2)

MT+NT(3,1) = 2

MT+NT(3,2) := sum(1, 1)

MT+NT(3,2) = 2

MT+NT(3,3) := sum(2, 0)

MT+NT(3,3) = 2

MT+NT(4,1) := sum(-1, 3)

MT+NT(4,1) = 2

MT+NT(4,2) := sum(0, 2)

MT+NT(4,2) = 2

MT+NT(4,3) := sum(1, 1)

MT+NT(4,3) = 2

MT+NT(5,1) := sum(-2, 4)

MT+NT(5,1) = 2

MT+NT(5,2) := sum(-1, 3)

MT+NT(5,2) = 2

MT+NT(5,3) := sum(0, 2)

MT+NT(5,3) = 2

left=[[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]];right=[[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]]

same value at every transposed coordinate

Reverse checks

• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Recompute MT+NT(5,3) with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute MT+NT(5,2) with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute MT+NT(5,1) with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute MT+NT(4,3) with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute MT+NT(4,2) with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute MT+NT(4,1) with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute MT+NT(3,3) with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute MT+NT(3,2) with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute MT+NT(3,1) with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute MT+NT(2,3) with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute MT+NT(2,2) with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute MT+NT(2,1) with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute M+N(3,5) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute M+N(3,4) with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute M+N(3,3) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute M+N(3,2) with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute M+N(2,5) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute M+N(2,4) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute M+N(2,3) with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute M+N(2,2) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute M+N(1,5) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute M+N(1,4) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute M+N(1,3) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute M+N(1,2) with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([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]])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.