[Math] How to find the answer of $\det[A^{-1}+4 \space adj(A)]$

Question

How to solve the following questoin?

If A is a $3 \times 3$ matrix and $\det(A)=2$, find $\det[A^{-1}+4 \space \rm adj(A)]$.

Can I do this?

\begin{eqnarray}
\\ \det[A^{-1}+4 \space \rm adj(A)] &=& \det(A^{-1})+4A^{-1}\det(A)\\
\\&=& \dfrac{1}{\det\left(A\right)}+4A^{-1}\det\left(A\right)\\
\\&=& \frac{1}{2}+8A^{-1}\\
\end{eqnarray}

but it seems like a wrong answer. What is the right answer?

There is one more question : Can I rewrite $\det(A^{-1})$ as $\det(A)^{-1}$?

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Thank you for your attention

Answer 1

Here is the answer from the solution paper.