Question 818221
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.)
<pre>
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
</pre>which translates into:
x = -1
y = 2
z = -4