# Real 2×2 Determinant Foundations

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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.



## 1. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Find det(M) exactly.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0001/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 2. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. Find det(M) exactly.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0002/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 3. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0003/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 4. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0004/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 5. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0005/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 6. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0006/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 7. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0007/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 8. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0008/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 9. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0009/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 10. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0010/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 11. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0011/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 12. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. Find det(M) exactly.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0012/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 13. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. Find det(M) exactly.

View answer and explanation2

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0013/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 14. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. Find det(M) exactly.

View answer and explanation18

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0014/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 15. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. Find det(M) exactly.

View answer and explanation36

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0015/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 16. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. Find det(M) exactly.

View answer and explanation56

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0016/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 17. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. Find det(M) exactly.

View answer and explanation70

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0017/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 18. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. Find det(M) exactly.

View answer and explanation94

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0018/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 19. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. Find det(M) exactly.

View answer and explanation120

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0019/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 20. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. Find det(M) exactly.

View answer and explanation148

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0020/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 21. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. Find det(M) exactly.

View answer and explanation170

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0021/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 22. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. Find det(M) exactly.

View answer and explanation202

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0022/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 23. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. Find det(M) exactly.

View answer and explanation236

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0023/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 24. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. Find det(M) exactly.

View answer and explanation272

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0024/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 25. Exact 2×2 Determinant

Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. Find det(M) exactly.

View answer and explanation302

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0025/)

Determinants of real 2×2 matrices · qf-determinant-two-by-two · candidate





## 26. Invertibility over the Real Field

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?

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0026/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 27. Invertibility over the Real Field

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?

View answer and explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0027/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 28. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0028/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 29. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0029/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 30. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0030/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 31. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0031/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 32. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0032/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 33. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0033/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 34. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0034/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 35. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0035/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 36. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0036/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 37. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0037/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 38. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0038/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 39. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0039/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 40. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0040/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 41. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0041/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 42. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0042/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 43. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0043/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 44. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0044/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 45. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0045/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 46. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0046/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 47. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0047/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 48. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0048/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 49. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0049/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 50. Invertibility over the Real Field

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?

View answer and explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0050/)

Determinants of real 2×2 matrices · qf-determinant-invertibility · candidate





## 51. Determinant of a Matrix Product

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.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0051/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 52. Determinant of a Matrix Product

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.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0052/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 53. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0053/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 54. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0054/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 55. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0055/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 56. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0056/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 57. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0057/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 58. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0058/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 59. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0059/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 60. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0060/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 61. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0061/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 62. Determinant of a Matrix Product

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.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0062/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 63. Determinant of a Matrix Product

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.

View answer and explanation52

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0063/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 64. Determinant of a Matrix Product

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.

View answer and explanation522

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0064/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 65. Determinant of a Matrix Product

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.

View answer and explanation1152

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0065/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 66. Determinant of a Matrix Product

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.

View answer and explanation1792

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0066/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 67. Determinant of a Matrix Product

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.

View answer and explanation2450

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0067/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 68. Determinant of a Matrix Product

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.

View answer and explanation3572

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0068/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 69. Determinant of a Matrix Product

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.

View answer and explanation4560

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0069/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 70. Determinant of a Matrix Product

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.

View answer and explanation6068

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0070/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 71. Determinant of a Matrix Product

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.

View answer and explanation7480

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0071/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 72. Determinant of a Matrix Product

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.

View answer and explanation8888

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0072/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 73. Determinant of a Matrix Product

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.

View answer and explanation11092

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0073/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 74. Determinant of a Matrix Product

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.

View answer and explanation13600

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0074/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 75. Determinant of a Matrix Product

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.

View answer and explanation15100

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0075/)

Determinants of real 2×2 matrices · qf-determinant-product · candidate





## 76. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. For c=0, find det(cM), with every entry multiplied by c.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0076/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 77. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. For c=-1, find det(cM), with every entry multiplied by c.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0077/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 78. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0078/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 79. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0079/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 80. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0080/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 81. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. For c=-2, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0081/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 82. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0082/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 83. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0083/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 84. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0084/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 85. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. For c=-2, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0085/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 86. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0086/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 87. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation-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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0087/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 88. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation2

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0088/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 89. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. For c=-2, find det(cM), with every entry multiplied by c.

View answer and explanation72

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0089/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 90. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation324

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0090/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 91. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation896

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0091/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 92. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation70

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0092/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 93. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. For c=-2, find det(cM), with every entry multiplied by c.

View answer and explanation376

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0093/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 94. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation1080

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0094/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 95. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation2368

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0095/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 96. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation170

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0096/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 97. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. For c=-2, find det(cM), with every entry multiplied by c.

View answer and explanation808

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0097/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 98. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. For c=3, find det(cM), with every entry multiplied by c.

View answer and explanation2124

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0098/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 99. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. For c=-4, find det(cM), with every entry multiplied by c.

View answer and explanation4352

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0099/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate





## 100. 2×2 Determinant after Scalar Multiplication

Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. For c=1, find det(cM), with every entry multiplied by c.

View answer and explanation302

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](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r03-determinant-0100/)

Determinants of real 2×2 matrices · qf-determinant-scalar · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r03-determinant-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261006-advance-20261007-r03-determinant-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r03-determinant-foundations/validation-report.json)