# Question 7610: Candidate Transpose Shape before Entrywise Residuals [Return to catalogue page 39](https://ocme.evemisslab.com/en/questions/page/39/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r02-matrix-foundations/#q-20261012-advance-20261016-r02-matrix-0085) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0014-07610 · q-20261012-advance-20261016-r02-matrix-0085 ## 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. 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). - 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. 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). ## 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](https://ocme.evemisslab.com/en/mko/mko-matrix/) ## Step-by-step derivation S001 · impliesBefore M=[[-3, 1], [2, -1]];shape=[2, 2] After external indices start at1;transpose is not inverse Conditions: none First confirm a nonempty rectangular shape and complete integer entries.Transposition is defined for rectangular matrices without invertibility and is not matrix inversion. Step identifiers and dependenciesRule: rectangular_matrix_scope · Uses: G001 S002 · impliesBefore M transpose:source(1,1)=-3 After target(1,1)=-3 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S001 S003 · impliesBefore M transpose:source(1,2)=1 After target(2,1)=1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S002 S004 · impliesBefore M transpose:source(2,1)=2 After target(1,2)=2 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S003 S005 · impliesBefore M transpose:source(2,2)=-1 After target(2,2)=-1 Conditions: none Transposition swaps row and column positions without changing the value.All displayed positions here are 1-based. Step identifiers and dependenciesRule: transpose_coordinate_swap · Uses: G001, S004 S006 · impliesBefore candidate shape=[2, 2];required transpose shape=[2, 2] After qualified=Yes Conditions: none The candidate must have the swapped row/column sizes of M.A shape mismatch forbids constructing an entrywise residual. Step identifiers and dependenciesRule: candidate_matrix_shape_before_difference · Uses: G001, S005 S007 · calculationBefore candidate-MT(1,1) := sub(-2, -3) After candidate-MT(1,1) = 1 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore candidate-MT(1,2) := sub(2, 2) After candidate-MT(1,2) = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore candidate-MT(2,1) := sub(1, 1) After candidate-MT(2,1) = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore candidate-MT(2,2) := sub(-1, -1) After candidate-MT(2,2) = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore qualified=Yes;residual/status=[[1, 0], [0, 0]] After equal=No Conditions: none With matching shapes every entry residual must be zero.Mismatched shapes fail a prerequisite and must not be assigned a fabricated zero residual matrix. Step identifiers and dependenciesRule: candidate_matrix_all_entries · Uses: G001, S010 ## Reverse checks - 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 — Recompute candidate-MT(2,2) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute candidate-MT(2,1) with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute candidate-MT(1,2) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute candidate-MT(1,1) with exact rational arithmetic, not rounded decimals. - 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, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2216c684ebb63f959c528720469d2969048c6e6c0449c7ddd866dc27e9c98f11 - candidate record SHA-256: 42e136a1d88464f3b1547e0d198353b545b007bcc3b1f6efcb783734f9197e7b - semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77 - question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r02-matrix-0085.json)