Elementary Linear Algebra 7th Edition

$$\operatorname{adj}(A)=\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1 \\ -1&2&-1&-1\\ 2&-1&-1&-1 \end {array} \right] .$$ $$A^{-1}= -\frac{1}{3}\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1 \\ -1&2&-1&-1\\ 2&-1&-1&-1 \end {array} \right].$$
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 $$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{3}\left[ \begin {array}{cccc} -1&-1&-1&2\\ -1&-1&2&-1 \\ -1&2&-1&-1\\ 2&-1&-1&-1 \end {array} \right].$$