There are several matrix-based methods for solving systems of equations like this. So I'm not sure what is meant by "the matrix method". I hope the following seems familiar. (Note: algebra.com does not do the brackets around matrices well. So I am going to show just the array of numbers.)
Initial matrix.
1 2 -3 15
2 -3 1 -12
-1 1 -3 15
Add -2 times row 1 to row 2
1 2 -3 15
0 -7 7 -42
-1 1 -3 15
Add 1 times row 1 to row 3
1 2 -3 15
0 -7 7 -42
0 3 -6 30
Replaced row 2 with -1/7 times row 2.
1 2 -3 15
0 1 -1 6
0 3 -6 30
Add -2 times row 2 to row 1
1 0 -1 3
0 1 -1 6
0 3 -6 30
Add -3 times row 2 to row 3.
1 0 -1 3
0 1 -1 6
0 0 -3 12
Replaced row 3 with -1/3 times row 3.
1 0 -1 3
0 1 -1 6
0 0 1 -4
Add 1 times row 3 to row 1.
1 0 0 -1
0 1 -1 6
0 0 1 -4
Add 1 times row 3 to row 2.
1 0 0 -1
0 1 0 2
0 0 1 -4
which translates into:
x = -1
y = 2
z = -4