Answer
$$\operatorname{adj}(A)=\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1
\\ -1&2&-1&-1\\ 2&-1&-1&-1
\end {array} \right]
.$$
$$A^{-1}= -\frac{1}{3}\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1
\\ -1&2&-1&-1\\ 2&-1&-1&-1
\end {array} \right].$$
Work Step by Step
The matrix is given by $A=\left[ \begin {array}{cccc} 1&1&1&0\\ 1&1&0&1
\\ 1&0&1&1\\ 0&1&1&1\end {array}
\right]
.$
To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows
$$\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1
\\ -1&2&-1&-1\\ 2&-1&-1&-1
\end {array} \right]
.$$
Now, the adjoint of $A$ is
$$\operatorname{adj}(A)=\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1
\\ -1&2&-1&-1\\ 2&-1&-1&-1
\end {array} \right]
.$$
To find $A^{-1}$, we have to calculate $\det(A)$ which is given by
$$\det(A)=-3.$$
Finally, we have
$$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{3}\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1
\\ -1&2&-1&-1\\ 2&-1&-1&-1
\end {array} \right].$$
