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 $-256/125.$

You have from wikipedia or maybe your book somewhere derived that: $${\bf A}adj({\bf A}) = det({\bf A}){\bf I}$$
which is also true for the inverse matrix if it exists:
$${\bf A}^{-1}adj({\bf A}^{-1}) = det({\bf A}^{-1}){\bf I}$$
Multiplying both sides $\bf A$ gives $adj({\bf A}^{-1}) = det({\bf A}^{-1}){\bf A}$
Plug that adj expression in and maybe you can take it from there.
