Question 7834: Signed Determinant, Both Ranks, and System Class

Original question and givens

Over R, rows=[[1, 1, -4], [0, 1, 2]], 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=[[1, 1, -4], [0, 1, 2]], 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

Step-by-step derivation

rows=[[1, 1, -4], [0, 1, 2]]

each row [a,b,c] means a*x+b*y=c over R

before ae := mul(1, 1)

before ae = 1

before bd := mul(1, 0)

before bd = 0

before D := sub(1, 0)

before D = 1

before af := mul(1, 2)

before af = 2

before cd := mul(-4, 0)

before cd = 0

before minor_x_rhs := sub(2, 0)

before minor_x_rhs = 2

before bf := mul(1, 2)

before bf = 2

before ce := mul(-4, 1)

before ce = -4

before minor_y_rhs := sub(2, -4)

before minor_y_rhs = 6

phase=before;rows=[[1, 1, -4], [0, 1, 2]];2x2 minors=[1, 2, 6]

rankA=2;rankAb=2

before x numerator := sub(0, 6)

before x numerator = -6

before x solution := div(-6, 1)

before x solution = -6

before y solution := div(2, 1)

before y solution = 2

before solution row0 ax := mul(1, -6)

before solution row0 ax = -6

before solution row0 by := mul(1, 2)

before solution row0 by = 2

before solution row0 lhs := sum(-6, 2)

before solution row0 lhs = -4

before solution row0 residual := sub(-4, -4)

before solution row0 residual = 0

before solution row1 ax := mul(0, -6)

before solution row1 ax = 0

before solution row1 by := mul(1, 2)

before solution row1 by = 2

before solution row1 lhs := sum(0, 2)

before solution row1 lhs = 2

before solution row1 residual := sub(2, 2)

before solution row1 residual = 0

phase=before;D=1;D*x=-6;D*y=2

solution=[-6/1, 2/1];row residuals=[0, 0]

phase=before;rankA=2;rankAb=2;unknowns=2

class=unique solution;solution/status=[-6/1, 2/1]

Reverse checks

• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute before solution row1 residual with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute before solution row1 lhs with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute before solution row1 by with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute before solution row1 ax with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute before solution row0 residual with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute before solution row0 lhs with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute before solution row0 by with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute before solution row0 ax with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute before y solution with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute before x solution with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute before x numerator with exact rational arithmetic, not rounded decimals.
• 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

(1, 2, 2, unique-solution, [-6/1, 2/1])

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.