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Question 7827: Signed Determinant, Both Ranks, and System Class

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0014-07827 · q-20261012-advance-20261016-r05-system-linear-equations-0027

Original question and givens

Over R, rows=[[0, 0, 1], [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, 1], [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.

Step-by-step derivation

S001 · implies

Before

rows=[[0, 0, 1], [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 dependencies

Rule: two_equations_real_system_scope · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

before cd := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

before ce := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

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

After

rankA=0;rankAb=1

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 dependencies

Rule: system_rank_from_complete_minors · Uses: G001, S010

S012 · implies

Before

phase=before;rankA=0;rankAb=1;unknowns=2

After

class=no solution;solution/status=not applicable

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 dependencies

Rule: 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, 1, no-solution, not-applicable)

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: 3c9a6d3943c7e876ba82a1bd9632f66c176083e3c28acbcec73c37061a36f0b7
  • candidate record SHA-256: d691211581e063581e51221f75f18a9fa06bf3a772fb75eb03974cb594458ab3
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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