# Question 7826: Signed Determinant, Both Ranks, and System Class [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-0026) 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. R0014-07826 · q-20261012-advance-20261016-r05-system-linear-equations-0026 ## Original question and givens Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency. - G001 — Over R, rows=[[0, 0, 0], [0, 0, 0]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency. ## 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 ## Reverse checks - 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, infinitely-many-solutions, not-unique) 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: 20a64cb8455b74fb4ff58964fbd216217eeb6f9a349530d4f550a76bfe693468 - candidate record SHA-256: a11959013992cdb8bd5598c55ec4e4bd8a1dd164f5859bf5d76ec9a7a897a1d6 - 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-0026.json)