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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261006-advance-20261007-r03-determinant-foundations · 2026-10-06 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Find det(M) exactly.
0
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=0. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. Find det(M) exactly.
0
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=0. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Find det(M) exactly.
-1
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-1. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Find det(M) exactly.
-28
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-28. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. Find det(M) exactly.
-38
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-38. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. Find det(M) exactly.
-38
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-38. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. Find det(M) exactly.
-36
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-36. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. Find det(M) exactly.
-32
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-32. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Find det(M) exactly.
-34
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-34. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. Find det(M) exactly.
-26
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-26. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. Find det(M) exactly.
-16
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-16. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. Find det(M) exactly.
-4
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=-4. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. Find det(M) exactly.
2
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=2. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. Find det(M) exactly.
18
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=18. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. Find det(M) exactly.
36
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=36. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. Find det(M) exactly.
56
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=56. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. Find det(M) exactly.
70
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=70. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. Find det(M) exactly.
94
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=94. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. Find det(M) exactly.
120
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=120. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. Find det(M) exactly.
148
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=148. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. Find det(M) exactly.
170
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=170. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. Find det(M) exactly.
202
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=202. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. Find det(M) exactly.
236
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=236. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. Find det(M) exactly.
272
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=272. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. Find det(M) exactly.
302
Multiply the main-diagonal entries and subtract the other diagonal product: ad−bc=302. Negative, zero, and positive determinants are all valid outputs.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate
Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse?
No
The determinant is 0. Over R, invertibility is equivalent to a nonzero determinant, giving No. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?
No
The determinant is 0. Over R, invertibility is equivalent to a nonzero determinant, giving No. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -1. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -28. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -38. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -38. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -36. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -32. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -34. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -26. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -16. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is -4. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 2. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 18. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 36. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 56. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 70. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 94. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 120. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 148. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 170. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 202. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 236. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 272. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. Is M invertible over the real field, without requiring an integer-entry inverse?
Yes
The determinant is 302. Over R, invertibility is equivalent to a nonzero determinant, giving Yes. A negative determinant may be nonzero; this criterion is not transferred unchanged to the integer ring.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate
Over R, A=[[0, 0], [0, 0]] and B=[[1, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
0
The determinants of A and B are 0 and 2, so det(AB)=0; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[1, 2], [2, 4]] and B=[[2, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
0
The determinants of A and B are 0 and 5, so det(AB)=0; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[0, 1], [1, 0]] and B=[[3, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-8
The determinants of A and B are -1 and 8, so det(AB)=-8; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-9, 4], [-2, 4]] and B=[[4, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-224
The determinants of A and B are -28 and 8, so det(AB)=-224; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-8, 1], [-2, 5]] and B=[[5, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-418
The determinants of A and B are -38 and 11, so det(AB)=-418; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-7, 2], [-2, 6]] and B=[[6, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-532
The determinants of A and B are -38 and 14, so det(AB)=-532; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-6, 3], [-2, 7]] and B=[[7, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-504
The determinants of A and B are -36 and 14, so det(AB)=-504; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-5, 4], [-2, 8]] and B=[[8, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-544
The determinants of A and B are -32 and 17, so det(AB)=-544; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-4, 1], [-2, 9]] and B=[[9, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-680
The determinants of A and B are -34 and 20, so det(AB)=-680; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-3, 2], [-2, 10]] and B=[[10, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-520
The determinants of A and B are -26 and 20, so det(AB)=-520; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-2, 3], [-2, 11]] and B=[[11, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-368
The determinants of A and B are -16 and 23, so det(AB)=-368; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[-1, 4], [-2, 12]] and B=[[12, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
-104
The determinants of A and B are -4 and 26, so det(AB)=-104; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[0, 1], [-2, 13]] and B=[[13, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
52
The determinants of A and B are 2 and 26, so det(AB)=52; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[1, 2], [-2, 14]] and B=[[14, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
522
The determinants of A and B are 18 and 29, so det(AB)=522; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[2, 3], [-2, 15]] and B=[[15, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
1152
The determinants of A and B are 36 and 32, so det(AB)=1152; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[3, 4], [-2, 16]] and B=[[16, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
1792
The determinants of A and B are 56 and 32, so det(AB)=1792; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[4, 1], [-2, 17]] and B=[[17, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
2450
The determinants of A and B are 70 and 35, so det(AB)=2450; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[5, 2], [-2, 18]] and B=[[18, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
3572
The determinants of A and B are 94 and 38, so det(AB)=3572; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[6, 3], [-2, 19]] and B=[[19, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
4560
The determinants of A and B are 120 and 38, so det(AB)=4560; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[7, 4], [-2, 20]] and B=[[20, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
6068
The determinants of A and B are 148 and 41, so det(AB)=6068; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[8, 1], [-2, 21]] and B=[[21, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
7480
The determinants of A and B are 170 and 44, so det(AB)=7480; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[9, 2], [-2, 22]] and B=[[22, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
8888
The determinants of A and B are 202 and 44, so det(AB)=8888; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[10, 3], [-2, 23]] and B=[[23, -1], [1, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
11092
The determinants of A and B are 236 and 47, so det(AB)=11092; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[11, 4], [-2, 24]] and B=[[24, -1], [2, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
13600
The determinants of A and B are 272 and 50, so det(AB)=13600; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, A=[[12, 1], [-2, 25]] and B=[[25, -1], [0, 2]] are 2×2 matrices. Find det(AB), where AB is the usual row-by-column product.
15100
The determinants of A and B are 302 and 50, so det(AB)=15100; acceptance also recomputes the actual product. This is not entrywise multiplication and does not assert AB=BA.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-product · candidate
Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. For c=0, find det(cM), with every entry multiplied by c.
0
In dimension two, det(cM)=c²det(M), giving 0. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. For c=-1, find det(cM), with every entry multiplied by c.
0
In dimension two, det(cM)=c²det(M), giving 0. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. For c=3, find det(cM), with every entry multiplied by c.
-9
In dimension two, det(cM)=c²det(M), giving -9. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. For c=-4, find det(cM), with every entry multiplied by c.
-448
In dimension two, det(cM)=c²det(M), giving -448. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. For c=1, find det(cM), with every entry multiplied by c.
-38
In dimension two, det(cM)=c²det(M), giving -38. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. For c=-2, find det(cM), with every entry multiplied by c.
-152
In dimension two, det(cM)=c²det(M), giving -152. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. For c=3, find det(cM), with every entry multiplied by c.
-324
In dimension two, det(cM)=c²det(M), giving -324. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. For c=-4, find det(cM), with every entry multiplied by c.
-512
In dimension two, det(cM)=c²det(M), giving -512. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. For c=1, find det(cM), with every entry multiplied by c.
-34
In dimension two, det(cM)=c²det(M), giving -34. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. For c=-2, find det(cM), with every entry multiplied by c.
-104
In dimension two, det(cM)=c²det(M), giving -104. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. For c=3, find det(cM), with every entry multiplied by c.
-144
In dimension two, det(cM)=c²det(M), giving -144. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. For c=-4, find det(cM), with every entry multiplied by c.
-64
In dimension two, det(cM)=c²det(M), giving -64. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. For c=1, find det(cM), with every entry multiplied by c.
2
In dimension two, det(cM)=c²det(M), giving 2. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. For c=-2, find det(cM), with every entry multiplied by c.
72
In dimension two, det(cM)=c²det(M), giving 72. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. For c=3, find det(cM), with every entry multiplied by c.
324
In dimension two, det(cM)=c²det(M), giving 324. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. For c=-4, find det(cM), with every entry multiplied by c.
896
In dimension two, det(cM)=c²det(M), giving 896. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. For c=1, find det(cM), with every entry multiplied by c.
70
In dimension two, det(cM)=c²det(M), giving 70. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. For c=-2, find det(cM), with every entry multiplied by c.
376
In dimension two, det(cM)=c²det(M), giving 376. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. For c=3, find det(cM), with every entry multiplied by c.
1080
In dimension two, det(cM)=c²det(M), giving 1080. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. For c=-4, find det(cM), with every entry multiplied by c.
2368
In dimension two, det(cM)=c²det(M), giving 2368. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. For c=1, find det(cM), with every entry multiplied by c.
170
In dimension two, det(cM)=c²det(M), giving 170. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. For c=-2, find det(cM), with every entry multiplied by c.
808
In dimension two, det(cM)=c²det(M), giving 808. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. For c=3, find det(cM), with every entry multiplied by c.
2124
In dimension two, det(cM)=c²det(M), giving 2124. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. For c=-4, find det(cM), with every entry multiplied by c.
4352
In dimension two, det(cM)=c²det(M), giving 4352. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. For c=1, find det(cM), with every entry multiplied by c.
302
In dimension two, det(cM)=c²det(M), giving 302. A negative c still acts through its square; c=0 gives determinant zero.
Reviewed step-by-step derivation
Determinants of real 2×2 matrices · qf-determinant-scalar · candidate
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