Question 7864: Reversible Row Addition and Before/After Classification

Original question and givens

Over R, rows=[[1, 0, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; 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, 0, 1], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=0; 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

rows=[[1, 0, 1], [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(0, 0)

before bd = 0

before D := sub(1, 0)

before D = 1

before af := mul(1, 2)

before af = 2

before cd := mul(1, 0)

before cd = 0

before minor_x_rhs := sub(2, 0)

before minor_x_rhs = 2

before bf := mul(0, 2)

before bf = 0

before ce := mul(1, 1)

before ce = 1

before minor_y_rhs := sub(0, 1)

before minor_y_rhs = -1

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

rankA=2;rankAb=2

before x numerator := sub(0, -1)

before x numerator = 1

before x solution := div(1, 1)

before x solution = 1

before y solution := div(2, 1)

before y solution = 2

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

before solution row0 ax = 1

before solution row0 by := mul(0, 2)

before solution row0 by = 0

before solution row0 lhs := sum(1, 0)

before solution row0 lhs = 1

before solution row0 residual := sub(1, 1)

before solution row0 residual = 0

before solution row1 ax := mul(0, 1)

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=1;D*y=2

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

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

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

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

t row1[0] = 0

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

t row1[1] = 0

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

t row1[2] = 0

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

row2+t row1[0] = 0

row2+t row1[1] := sum(1, 0)

row2+t row1[1] = 1

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

row2+t row1[2] = 2

operation=system-eliminate-row;t=0;old rows=[[1, 0, 1], [0, 1, 2]]

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

after ae := mul(1, 1)

after ae = 1

after bd := mul(0, 0)

after bd = 0

after D := sub(1, 0)

after D = 1

after af := mul(1, 2)

after af = 2

after cd := mul(1, 0)

after cd = 0

after minor_x_rhs := sub(2, 0)

after minor_x_rhs = 2

after bf := mul(0, 2)

after bf = 0

after ce := mul(1, 1)

after ce = 1

after minor_y_rhs := sub(0, 1)

after minor_y_rhs = -1

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

rankA=2;rankAb=2

after x numerator := sub(0, -1)

after x numerator = 1

after x solution := div(1, 1)

after x solution = 1

after y solution := div(2, 1)

after y solution = 2

after solution row0 ax := mul(1, 1)

after solution row0 ax = 1

after solution row0 by := mul(0, 2)

after solution row0 by = 0

after solution row0 lhs := sum(1, 0)

after solution row0 lhs = 1

after solution row0 residual := sub(1, 1)

after solution row0 residual = 0

after solution row1 ax := mul(0, 1)

after solution row1 ax = 0

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

after solution row1 by = 2

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

after solution row1 lhs = 2

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

after solution row1 residual = 0

phase=after;D=1;D*x=1;D*y=2

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

phase=after;rankA=2;rankAb=2;unknowns=2

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

Reverse checks

• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Recompute after solution row1 residual with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute after solution row1 lhs with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute after solution row1 by with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute after solution row1 ax with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute after solution row0 residual with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute after solution row0 lhs with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute after solution row0 by with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute after solution row0 ax with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute after y solution with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute after x solution with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute after x numerator with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute after minor_y_rhs with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute after ce with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute after bf with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute after minor_x_rhs with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute after cd with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute after af with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute after D with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute after bd with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute after ae with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute row2+t row1[2] with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute row2+t row1[1] with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute row2+t row1[0] with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute t row1[2] with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute t row1[1] with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute t row1[0] with exact rational arithmetic, not rounded decimals.
• 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, 0, 1], [0, 1, 2]], [1, 2, 2, unique-solution, [1/1, 2/1]], [1, 2, 2, unique-solution, [1/1, 2/1]], 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.