Which of the following statements are true?
I. A square matrix is invertible exactly when its determinant is not 0.
II. If $A$ is singular, then $A(\operatorname{adj} A) = O$ .
III. $A^{-1} = \frac{1}{|A|}\operatorname{adj} A$ for every square matrix $A$ .
IV. $|A^{-1}| = |A|$ for every invertible matrix $A$ .

I, II and III onlyCorrect answerYour answerI onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answerI and II onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

I is the test for an inverse. II: $A(\operatorname{adj} A) = |A|I$ , and $|A| = 0$ makes it $O$ .
III fails when $|A| = 0$ , where there is no inverse to find; IV is false because $|A^{-1}| = \frac{1}{|A|}$ .

Concept: Facts about the inverse
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe adjoint and the inverse

Practise this chapter →

Ten questions from Determinants, Adjoint & Inverse, at your own pace.

Explanation