Question

Let , $A$ is a non-singular idempotent matrix of order $3\times 3$ and $B$ is an involuntary matrix of same order. If $\text{det}\left(A\right)\neq det \left(B\right) \, $ , then $\text{det}\left(A^{- 5} B^{- 2}\right)=$

Answer 1

$\text{det}\left(A\right)=1, \, \text{det} \left(B\right)=-1$
$\Rightarrow \text{det} \left(A^{- 1}\right)=1, \, \text{det} ⁡ \left(B^{- 1}\right)=-1$