Answer
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc}4&0 \\0&1
\end {array} \right].$$
$$A^{-1}= -\frac{1}{4}\left[ \begin {array}{ccc}4&0 \\0&1
\end {array} \right].$$
Work Step by Step
The matrix is given by $A=\left[ \begin {array}{ccc}-1&0 \\0&4
\end {array} \right].$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows
$$\left[ \begin {array}{ccc}4&0 \\0&1
\end {array} \right].$$
Now, the adjoint of $A$ is
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc}4&0 \\0&1
\end {array} \right].$$
To find $A^{-1}$, we have to calculate $\det(A)$ which is given by
$$\det(A)=-4.$$
Finally, we have
$$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{4}\left[ \begin {array}{ccc}4&0 \\0&1
\end {array} \right].$$