I guess you mean solve the system using matrices.
First, evaluate the determinant (of the variable coefficients) Δ=
| 1 -1 3 |
| 4 1 2 | = 1(2-4)+1(8-6)+3(8-3)=-2+2+15=15
| 3 2 2 |
Next evaluate the determinant associated with the variables x, y, z, replacing the variable column by the column of constants:
Δx=
| 13 -1 3 |
| 17 1 2 | = 13(2-4)+1(34-2)+3(34-1)=-26+32+99=105
| 1 2 2 |
x=Δx/Δ=105/15=7.
Δy=
| 1 13 3 |
| 4 17 2 | = 1(34-2)-13(8-6)+3(4-51)=32-26-141=-135
| 3 1 2 |
y=Δy/Δ=-135/15=-9.
Δz=
| 1 -1 13 |
| 4 1 17 | = 1(1-34)+1(4-51)+13(8-3)=-33-47+65=-15
| 3 2 1 |
z=Δz/Δ=-15/15=-1.
SOLUTION: x=7, y=-9, z=-1.