The determinant of the matrix given below is

The given matrix is:

[ 0   1   0   2 ]
[ -1  1   1   3 ]
[ 0   0   0   1 ]
[ 1  -2   0   1 ]

Expand the determinant along the third row. The only nonzero entry in the third row is 1 in column 4.

The cofactor sign for position (3,4) is (-1)^(3+4) = -1.

Now delete row 3 and column 4. The remaining 3 x 3 matrix is:

[ 0   1   0 ]
[ -1  1   1 ]
[ 1  -2   0 ]

Its determinant is 1. Therefore the determinant of the original matrix is:

-1 x 1 = -1.

So the determinant is -1.