Question 7341: Both Matrix Product Orders, Determinants, and Equality

Original question and givens

Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).

• G001 — Use signed determinants of real 2×2 matrices with integer entries; ordinary matrix products are row-by-column, not entrywise products. Real-field invertibility is not integer-ring invertibility. M=[[3, 1], [0, 1]], N=[[1, 0], [2, 1]]. Give (MN, NM, D(MN), D(NM), D(M)D(N), are MN and NM equal?).

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-determinant

Step-by-step derivation

[[3, 1], [0, 1]]

real2×2 integer-entry slice;determinants are signed

M diagonal product := mul(3, 1)

M diagonal product = 3

M off-diagonal product := mul(1, 0)

M off-diagonal product = 0

M signed determinant := sub(3, 0)

M signed determinant = 3

MN entry0,0 term0 := mul(3, 1)

MN entry0,0 term0 = 3

MN entry0,0 term1 := mul(1, 2)

MN entry0,0 term1 = 2

MN entry0,0 := sum(3, 2)

MN entry0,0 = 5

product=MN;row=0;column=0;terms=[3, 2]

entry=5

MN entry0,1 term0 := mul(3, 0)

MN entry0,1 term0 = 0

MN entry0,1 term1 := mul(1, 1)

MN entry0,1 term1 = 1

MN entry0,1 := sum(0, 1)

MN entry0,1 = 1

product=MN;row=0;column=1;terms=[0, 1]

entry=1

MN entry1,0 term0 := mul(0, 1)

MN entry1,0 term0 = 0

MN entry1,0 term1 := mul(1, 2)

MN entry1,0 term1 = 2

MN entry1,0 := sum(0, 2)

MN entry1,0 = 2

product=MN;row=1;column=0;terms=[0, 2]

entry=2

MN entry1,1 term0 := mul(0, 0)

MN entry1,1 term0 = 0

MN entry1,1 term1 := mul(1, 1)

MN entry1,1 term1 = 1

MN entry1,1 := sum(0, 1)

MN entry1,1 = 1

product=MN;row=1;column=1;terms=[0, 1]

entry=1

NM entry0,0 term0 := mul(1, 3)

NM entry0,0 term0 = 3

NM entry0,0 term1 := mul(0, 0)

NM entry0,0 term1 = 0

NM entry0,0 := sum(3, 0)

NM entry0,0 = 3

product=NM;row=0;column=0;terms=[3, 0]

entry=3

NM entry0,1 term0 := mul(1, 1)

NM entry0,1 term0 = 1

NM entry0,1 term1 := mul(0, 1)

NM entry0,1 term1 = 0

NM entry0,1 := sum(1, 0)

NM entry0,1 = 1

product=NM;row=0;column=1;terms=[1, 0]

entry=1

NM entry1,0 term0 := mul(2, 3)

NM entry1,0 term0 = 6

NM entry1,0 term1 := mul(1, 0)

NM entry1,0 term1 = 0

NM entry1,0 := sum(6, 0)

NM entry1,0 = 6

product=NM;row=1;column=0;terms=[6, 0]

entry=6

NM entry1,1 term0 := mul(2, 1)

NM entry1,1 term0 = 2

NM entry1,1 term1 := mul(1, 1)

NM entry1,1 term1 = 1

NM entry1,1 := sum(2, 1)

NM entry1,1 = 3

product=NM;row=1;column=1;terms=[2, 1]

entry=3

MN diagonal product := mul(5, 1)

MN diagonal product = 5

MN off-diagonal product := mul(1, 2)

MN off-diagonal product = 2

MN signed determinant := sub(5, 2)

MN signed determinant = 3

NM diagonal product := mul(3, 3)

NM diagonal product = 9

NM off-diagonal product := mul(1, 6)

NM off-diagonal product = 6

NM signed determinant := sub(9, 6)

NM signed determinant = 3

N diagonal product := mul(1, 1)

N diagonal product = 1

N off-diagonal product := mul(0, 2)

N off-diagonal product = 0

N signed determinant := sub(1, 0)

N signed determinant = 1

product of original determinants := mul(3, 1)

product of original determinants = 3

MN=[[5, 1], [2, 1]];NM=[[3, 1], [6, 3]];determinants=(3,3)

matrices equal=No;D(M)D(N)=3

Reverse checks

• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute product of original determinants with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute N signed determinant with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute N off-diagonal product with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute N diagonal product with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute NM signed determinant with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute NM off-diagonal product with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute NM diagonal product with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute MN signed determinant with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute MN off-diagonal product with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute MN diagonal product with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute NM entry1,1 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute NM entry1,1 term1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute NM entry1,1 term0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute NM entry1,0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute NM entry1,0 term1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute NM entry1,0 term0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute NM entry0,1 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute NM entry0,1 term1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute NM entry0,1 term0 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 — Recompute NM entry0,0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute NM entry0,0 term1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute NM entry0,0 term0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute MN entry1,1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute MN entry1,1 term1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute MN entry1,1 term0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute MN entry1,0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute MN entry1,0 term1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute MN entry1,0 term0 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 — Recompute MN entry0,1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute MN entry0,1 term1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute MN entry0,1 term0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute MN entry0,0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute MN entry0,0 term1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute MN entry0,0 term0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute M signed determinant with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute M off-diagonal product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute M diagonal product 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

([[5, 1], [2, 1]], [[3, 1], [6, 3]], 3, 3, 3, No)

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.