Given determinant and dimension, find $\det(\operatorname{adj}(A))$?

Question

$\newcommand{\adj}{\operatorname{adj}}$

If $\det(A)=2$ and $n=5$ find $\det(\adj(A))$

My Attempt;
By Cramers Rule we know $$A \adj(A)=\det(A) \ \ I_n=\adj(A) A$$
So $$A \adj(A)= 2I_n$$
now $$\det(A \adj(A))=\det(A) \adj(A)= 2\adj(A)= \det(2I_n) =32 $$
and $\det(\adj(A))=16$
Is this correct?

Answer 1

I guess your last line should be:$\newcommand{\adj}{\operatorname{adj}}$ \begin{equation}\det(A \adj(A))=\det(A) \det(\adj(A))= 2\det(\adj(A))= \det(2I_n) =32. \end{equation} Apart from that it's correct.