$$\begin{bmatrix}
0 &1 &0 &2 \\
-1& 1& 1& 3\\
0&0 &0 & 1\\
1& -2& 0& 1
\end{bmatrix}$$
Reduce this matrix into Upper Triangular matrix using row and column transformations:
$R1\leftrightarrow R2$
$R3\leftrightarrow R4$
$$\begin{bmatrix}
-1& 1& 1& 3\\
0 &1 &0 &2 \\
1& -2& 0& 1\\
0&0 &0 & 1
\end{bmatrix}$$
$R3\leftarrow R1 + R3$
$R3\leftarrow R2 - R3$
Resulting Upper Triangular matrix will be:
$$\begin{bmatrix}
-1& 1& 1& 3\\
0 &1 &0 &2 \\
0& 0& 1& 2\\
0&0 &0 & 1
\end{bmatrix}$$
Determinant will be product of diagonal elements = $-1*1*1*1=-1$
Hence, A is correct option!