Question 7852: Reversible Row Addition and Before/After Classification

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0014-07852 · q-20261012-advance-20261016-r05-system-linear-equations-0052

Original question and givens

Over R, rows=[[1, 1, 1], [2, 2, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; replace only row2 by row2+t*row1, 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, invertible-row-operation eligibility).

G001 — Over R, rows=[[1, 1, 1], [2, 2, 2]], where each [a,b,c] means a*x+b*y=c. t=-2; replace only row2 by row2+t*row1, 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, invertible-row-operation eligibility).

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

S001 · implies

Before

rows=[[1, 1, 1], [2, 2, 2]]

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(1, 2)

After

before ae = 2

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(1, 2)

After

before bd = 2

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(2, 2)

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(1, 2)

After

before af = 2

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, 2)

After

before cd = 2

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(2, 2)

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(1, 2)

After

before bf = 2

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, 2)

After

before ce = 2

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(2, 2)

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=[[1, 1, 1], [2, 2, 2]];2x2 minors=[0, 0, 0]

After

rankA=1;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=1;rankAb=1;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 dependencies

Rule: system_consistency_and_free_variables · Uses: G001, S011

S013 · calculation

Before

t row1[0] := mul(-2, 1)

After

t row1[0] = -2

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, S012

S014 · calculation

Before

t row1[1] := mul(-2, 1)

After

t row1[1] = -2

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, S013

S015 · calculation

Before

t row1[2] := mul(-2, 1)

After

t row1[2] = -2

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, S014

S016 · calculation

Before

row2+t row1[0] := sum(2, -2)

After

row2+t row1[0] = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

row2+t row1[1] := sum(2, -2)

After

row2+t row1[1] = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

row2+t row1[2] := sum(2, -2)

After

row2+t row1[2] = 0

Conditions: none

Add the already aligned component or position,without mixing coordinates.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

operation=system-eliminate-row;t=-2;old rows=[[1, 1, 1], [2, 2, 2]]

After

new rows=[[1, 1, 1], [0, 0, 0]];invertible=Yes;inverse=row2 -> row2-(-2)*row1

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 dependencies

Rule: row_operation_inverse_guard · Uses: G001, S018

S020 · calculation

Before

after ae := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

after bd := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

after af := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

after cd := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

after bf := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

after ce := mul(1, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

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

After

rankA=1;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, S028

S030 · implies

Before

phase=after;rankA=1;rankAb=1;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 dependencies

Rule: system_consistency_and_free_variables · Uses: G001, S029

Reverse checks

S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute after minor_y_rhs with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute after ce with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute after bf with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute after minor_x_rhs with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute after cd with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute after af with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute after D with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute after bd with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute after ae with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Recompute row2+t row1[2] with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute row2+t row1[1] with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute row2+t row1[0] with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute t row1[2] with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute t row1[1] with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute t 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

([[1, 1, 1], [0, 0, 0]], [0, 1, 1, infinitely-many-solutions, not-unique], [0, 1, 1, infinitely-many-solutions, not-unique], Yes)

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: 616e667230e4f8a745200f3989b6beb60ac28721eccc09651007a8860af65250
candidate record SHA-256: 69be4002ade8f294d6869003965c768d656f428f99ea9c4904b13b69b606422f
semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
Lean: NOT_RUN · human review: NOT_PERFORMED

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