# Question 33198: Transpose of a Rectangular Matrix [Return to catalogue page 166](https://ocme.evemisslab.com/en/questions/page/166/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r01-matrix-foundations/#q-20261006-advance-20261007-r01-matrix-0098) 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. R0065-33198 · q-20261006-advance-20261007-r01-matrix-0098 ## Original question and givens The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows. - G001 — The real matrix M=[[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]] is displayed as nested rows. Find Mᵀ as a nested list of rows. ## 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 source roles = [qf-matrix-transpose, matrix_transpose, reals, []] After derived = [[nested_ROW_matrix]] Conditions: none MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status. Step identifiers and dependenciesRule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001 S002 · impliesBefore source roles = [M ORIGINAL, [[10, 9, 8], [12, 11, 10], [14, 13, 12], [16, 15, 14], [18, 17, 16]]] After derived = [5, 3] Conditions: none THIS bank allows1..5 by1..5 matrices,not the R64 2×2 restriction.Type EVERY raw SAFE cell,including unused shape/entry cells.Keep original row order. Step identifiers and dependenciesRule: Matrix_RAW_rectangular_1to5_ALL_SAFE_entries_even_unused · Uses: G001, S001 S003 · impliesBefore source roles = [1, 1, 10] After derived = [1, 1, 10] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S002 S004 · impliesBefore source roles = [2, 1, 12] After derived = [1, 2, 12] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S003 S005 · impliesBefore source roles = [3, 1, 14] After derived = [1, 3, 14] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S004 S006 · impliesBefore source roles = [4, 1, 16] After derived = [1, 4, 16] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S005 S007 · impliesBefore source roles = [5, 1, 18] After derived = [1, 5, 18] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S006 S008 · impliesBefore source roles = [1, 2, 9] After derived = [2, 1, 9] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S007 S009 · impliesBefore source roles = [2, 2, 11] After derived = [2, 2, 11] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S008 S010 · impliesBefore source roles = [3, 2, 13] After derived = [2, 3, 13] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S009 S011 · impliesBefore source roles = [4, 2, 15] After derived = [2, 4, 15] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S010 S012 · impliesBefore source roles = [5, 2, 17] After derived = [2, 5, 17] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S011 S013 · impliesBefore source roles = [1, 3, 8] After derived = [3, 1, 8] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S012 S014 · impliesBefore source roles = [2, 3, 10] After derived = [3, 2, 10] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S013 S015 · impliesBefore source roles = [3, 3, 12] After derived = [3, 3, 12] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S014 S016 · impliesBefore source roles = [4, 3, 14] After derived = [3, 4, 14] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S015 S017 · impliesBefore source roles = [5, 3, 16] After derived = [3, 5, 16] Conditions: none Move EVERY original(i,j)to(j,i),not row reversal or matrix inversion.Rectangular dimensions exchange. Step identifiers and dependenciesRule: Matrix_TRANSPOSE_EACH_original_i_j_moves_to_j_i · Uses: G001, S016 ## Reverse checks - S017 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S016 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S015 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S012 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S011 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S008 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S007 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. ## Result [[10, 12, 14, 16, 18], [9, 11, 13, 15, 17], [8, 10, 12, 14, 16]] 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: 7dbe2e06ed8a27bcdde4d2433a7499dcff6bbd0ecb4c70b0507c22d38dc5fa93 - candidate record SHA-256: 2f1af7e1fd526820d5dc4f738d0554bad0d4dfd7b30ad68c2380288e7d7bd35b - semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6 - question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r01-matrix-0098.json)