If $A$ & $B$ are $4\times 4$ matrices with $\det(A)=-5$ & $\det(B)=10$ then evaluate... If $A$ & $B$ are $4\times 4$ matrices with $\det(A)=-5 $ & $\det(B)=10$ then evaluate...
a) $\det\left(A+\operatorname{adj}\left(A^{-1}\right)\right)$
b) $\det(A+B)$
Yes, those are meant to be addition signs. I wouldn't be asking if it were multiplication. ANS for a) is $-256/125.$

Answer to a):
It holds that:
$$ A \cdot \operatorname{adj(A)} = \det (A) \cdot I .$$
Thus, we have:
$$ A^{-1}\cdot \operatorname{adj}\left(A^{-1}\right) = \det \left(A^{-1}\right) \cdot I.$$
However, $\det\left(A^{-1}\right) = -\frac{1}{5}$ and you need to apply the identity $\det(\lambda A) = \lambda^n \det A,$ where $A$ is a $n\times n$ matrix and $\lambda \in \mathbb R.$
