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Question

Mathematics Question on Invertible Matrices

If AA is a square matrix of order 3 , then | adj (adj⁡A2)∣\left(\operatorname{adj} A^{2}\right) \mid is equal to

A

∣A∣2|A |^{2}

B

∣A∣4|A|^{4}

C

∣A∣8|A|^{8}

D

∣A∣16|A|^{16}

Answer

∣A∣8|A|^{8}

Explanation

Solution

We know that,
∣adj⁡(adj⁡A)∣=∣A∣(n−1)2|\operatorname{adj}(\operatorname{adj} A)| =|A|^{(n-1)^{2}}
∣adj⁡(adj⁡A2)∣=∣A2∣(3−1)2\left|\operatorname{adj}\left(\operatorname{adj} A^{2}\right)\right| =\left|A^{2}\right|^{(3-1)^{2}}
=∣A2∣2  [∵n=3]=\left|A^{2}\right|^{2} \,\,[\because n=3]
=∣A2∣4=∣A∣8=\left|A^{2}\right|^{4}=|A|^{8}