Answer
3
Work Step by Step
Given $$\begin{array}{c}{A=\left[\begin{array}{ccc}
{0} & {1} & -1&1 \\
{-1} & {0} & 1&1 \\
1& -1 &0& 1\\
-1 &-1 & -1 &0\end{array}\right] }\end{array} $$
So, we get
$$A=\begin{vmatrix} {0} & {1} & -1&1 \\
{-1} & {0} & 1&1 \\
1& -1 &0& 1\\
-1 &-1 & -1 &0
\end{vmatrix} \xrightarrow{R_3+R_{1},R_4-R_1} \begin{vmatrix}
{-1} & {0} & 1&1 \\
{0} & {1} & -1&1 \\
0& -1 &1& 2\\
0 &-1 & 0 &-1
\end{vmatrix} \xrightarrow{R_{3}+R_{2},R_4+R_2 }\begin{vmatrix}
{-1} & {0} & 1&1 \\
{0} & {1} & -1&1 \\
0&4 0 &-1&0\\
0 &0& 0 &3
\end{vmatrix} $$
\begin{array}{l}{\text { Determinant of triangular matrix is the product of its diagonal elements, so: }} \\ {\text {det }A=-1 \cdot 1\cdot (-1)\cdot 3=3}\end{array}
