Given the adjoint $\mathrm{adj}(A)$, how do you find $\det(A)$ and $A^{-1}$? Given $\mathrm{adj}(A)$ where $A$ is an $n\times n$ matrix, how do you find the value of $\det(A)$ and $A^{-1}$?

Hint 
$$
\left( \begin{array}{ccc}
\text{det}(A) &  &  0\\ 
 & \ddots &  \\ 
0 &  & \text{det}(A)
\end{array} \right)=\text{det}(A) \cdot \text{I}_n = A \cdot \text{adj}(A)
$$
and 
$$
A^{-1}= \frac{1}{\text{det}(A)}\text{adj}(A)
$$
