Your matrix

        X1	X2	X3	X4	b
1	1	2	3	4	0
2	8	5	1	4	0
3	5	6	8	1	0
4	8	3	7	2	0

Write down the main matrix and find its determinant

        X1	X2	X3	X4
1	1	2	3	4
2	8	5	1	4
3	5	6	8	1
4	8	3	7	2

Δ = -1034    <<<---=== Since the determinant is not zero, the solution to the system is the zero vector.
                       Explicit calculations below confirm this fact.


Replace the 1st column of the main matrix with the solution vector and find its determinant

        X1	X2	X3	X4
1	0	2	3	4
2	0	5	1	4
3	0	6	8	1
4	0	3	7	2

Δ1 = 0


Replace the 2nd column of the main matrix with the solution vector and find its determinant

        X1	X2	X3	X4
1	1	0	3	4
2	8	0	1	4
3	5	0	8	1
4	8	0	7	2

Δ2 = 0


Replace the 3rd column of the main matrix with the solution vector and find its determinant

        X1	X2	X3	X4
1	1	2	0	4
2	8	5	0	4
3	5	6	0	1
4	8	3	0	2

Δ3 = 0


Replace the 4th column of the main matrix with the solution vector and find its determinant

        X1	X2	X3	X4
1	1	2	3	0
2	8	5	1	0
3	5	6	8	0
4	8	3	7	0

Δ4 = 0


x1 = Δ1 / Δ = 0 / (-1034) = 0
x2 = Δ2 / Δ = 0 / (-1034) = 0
x3 = Δ3 / Δ = 0 / (-1034) = 0
x4 = Δ4 / Δ = 0 / (-1034) = 0

Solution set:

x1 = 0
x2 = 0
x3 = 0
x4 = 0