If A is 3 x 3 matrix such that det(A) = 2. Then det(adj(adj(adj()adj A))) is

Question

If A is \(3 \times 3\) matrix such that \(\operatorname{det}(\mathrm{A})=2.\) Then \(\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))\) is

(1) \(2^{32}\)

(2) \(2^{16}\)

(3) \(2^8\)

(4) \(2^{12}\)

Answer 1

Correct option is (2) \(2^{16}\)   

\(\operatorname{det} \underbrace{(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}_{r \text { times }}=|A|^{(n-1)^r}\)  

\(\text { As }|A|=2 \)   

\(\therefore \operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adjA})))) =|A|^{(3-1)^4} \)   

\(=|A|^{2^4}=|A|^{16} \)

\(=2^{16}\)