Which of the following statements are true? I. Every minor of a 3 × 3 determinant is a 2 × 2 determinant. II. Aᵢⱼ = Mᵢⱼ whenever i + j is even. III. The elements of any column times their cofactors add up to Δ. IV. a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ = Δ.

Question 42

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Which of the following statements are true?
I. Every minor of a 3 × 3 determinant is a 2 × 2 determinant.
II. $A_{ij} = M_{ij}$ whenever $i + j$ is even.
III. The elements of any column times their cofactors add up to $\Delta$ .
IV. $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = \Delta$ .

All four: I, II, III and IVCorrect answerYour answerI, II and III onlyCorrect answerYour answerI and II onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer B

Correct answer: B

Deleting a row and a column of a 3 × 3 leaves a 2 × 2 (I); $(-1)^{i + j} = 1$ when $i + j$ is even (II); and any row or column expands to $\Delta$ (III).
IV pairs row 1's elements with row 2's cofactors, which always gives 0, so IV is false.

Concept: Facts about minors and cofactors
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsMinors and cofactors

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