# Question 7876: Row Scaling: Classification versus Operation Reversibility [Return to catalogue page 40](https://ocme.evemisslab.com/en/questions/page/40/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r05-system-linear-equations-foundations/#q-20261012-advance-20261016-r05-system-linear-equations-0076) 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. R0014-07876 · q-20261012-advance-20261016-r05-system-linear-equations-0076 ## Original question and givens Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal. - G001 — Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. t=0; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal. ## 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-system-linear-equations](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) ## Step-by-step derivation S001 · impliesBefore rows=[[0, 0, 0], [0, 0, 0]] After each row [a,b,c] means a*x+b*y=c over R Conditions: none Both equations must hold simultaneously.The right-hand constant is not a third unknown.Retain degenerate zero-coefficient rows rather than presuming a nonzero determinant. Step identifiers and dependenciesRule: two_equations_real_system_scope · Uses: G001 S002 · calculationBefore before ae := mul(0, 0) After before ae = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore before bd := mul(0, 0) After before bd = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore before D := sub(0, 0) After before D = 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, S003 S005 · calculationBefore before af := mul(0, 0) After before af = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore before cd := mul(0, 0) After before cd = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore before minor_x_rhs := sub(0, 0) After before minor_x_rhs = 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, S006 S008 · calculationBefore before bf := mul(0, 0) After before bf = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore before ce := mul(0, 0) After before ce = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore before minor_y_rhs := sub(0, 0) After before minor_y_rhs = 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 phase=before;rows=[[0, 0, 0], [0, 0, 0]];2x2 minors=[0, 0, 0] After rankA=0;rankAb=0 Conditions: none A nonzero coefficient 2×2 minor gives rank2;otherwise a nonzero coefficient gives rank1.The augmented matrix has three 2×2 minors;if all vanish,check its entries,including the right-hand constants. Step identifiers and dependenciesRule: system_rank_from_complete_minors · Uses: G001, S010 S012 · impliesBefore phase=before;rankA=0;rankAb=0;unknowns=2 After class=infinitely many solutions;solution/status=not unique Conditions: none Unequal ranks mean an inconsistent row and no solution.Equal rank below2 leaves a free variable and infinitely many real solutions.Only rank2 gives a unique solution.Never divide by D when D=0. Step identifiers and dependenciesRule: system_consistency_and_free_variables · Uses: G001, S011 S013 · calculationBefore scaled row1[0] := mul(0, 0) After scaled row1[0] = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore scaled row1[1] := mul(0, 0) After scaled row1[1] = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore scaled row1[2] := mul(0, 0) After scaled row1[2] = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore operation=system-row-scaling-solution;t=0;old rows=[[0, 0, 0], [0, 0, 0]] After new rows=[[0, 0, 0], [0, 0, 0]];invertible=No;inverse=not invertible Conditions: none A row operation includes the right-hand constant.Adding any t times the other row is undone by adding−t times it.Scaling can be divided back only for t≠0.At t=0 a redundant system may coincidentally keep its solution set,but the operation is still not invertible. Step identifiers and dependenciesRule: row_operation_inverse_guard · Uses: G001, S015 S017 · calculationBefore after ae := mul(0, 0) After after ae = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore after bd := mul(0, 0) After after bd = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore after D := sub(0, 0) After after D = 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, S018 S020 · calculationBefore after af := mul(0, 0) After after af = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore after cd := mul(0, 0) After after cd = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore after minor_x_rhs := sub(0, 0) After after minor_x_rhs = 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, S021 S023 · calculationBefore after bf := mul(0, 0) After after bf = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore after ce := mul(0, 0) After after ce = 0 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore after minor_y_rhs := sub(0, 0) After after minor_y_rhs = 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, S024 S026 · impliesBefore phase=after;rows=[[0, 0, 0], [0, 0, 0]];2x2 minors=[0, 0, 0] After rankA=0;rankAb=0 Conditions: none A nonzero coefficient 2×2 minor gives rank2;otherwise a nonzero coefficient gives rank1.The augmented matrix has three 2×2 minors;if all vanish,check its entries,including the right-hand constants. Step identifiers and dependenciesRule: system_rank_from_complete_minors · Uses: G001, S025 S027 · impliesBefore phase=after;rankA=0;rankAb=0;unknowns=2 After class=infinitely many solutions;solution/status=not unique Conditions: none Unequal ranks mean an inconsistent row and no solution.Equal rank below2 leaves a free variable and infinitely many real solutions.Only rank2 gives a unique solution.Never divide by D when D=0. Step identifiers and dependenciesRule: system_consistency_and_free_variables · Uses: G001, S026 ## Reverse checks - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute after minor_y_rhs with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute after ce with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute after bf with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute after minor_x_rhs with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute after cd with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute after af with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute after D with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute after bd with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute after ae with exact rational arithmetic, not rounded decimals. - 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 — Recompute scaled row1[2] with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute scaled row1[1] with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute scaled row1[0] with exact rational arithmetic, not rounded decimals. - 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 — Recompute before minor_y_rhs with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute before ce with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute before bf with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute before minor_x_rhs with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute before cd with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute before af with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute before D with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute before bd with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute before ae with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[0, 0, 0], [0, 0, 0]], [0, 0, 0, infinitely-many-solutions, not-unique], [0, 0, 0, infinitely-many-solutions, not-unique], No, Only nonzero scaling is an invertible row operation) 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: 72ee81c137586078ea2b35b32cbb83f903f8849a06974e4e6dbbba4e0b00147b - candidate record SHA-256: 1f3420bebe59907055f6d624ada90c1629e9fd408a8e885f060641eccc911144 - 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-r05-system-linear-equations-0076.json)