While preferences may vary, the procedure to solve a system using matrices is always the same. This is practical if you are not sure what algebraic method is better for the given situation. Nevertheless, if the determinant of the matrix is zero we cannot solve the system with it, but there can still be solutions. Let's see an example.
y - x = 1 & (I) 3y - 3x = 3 & (II)
In this case we can see that the equations represent coinciding lines, and therefore, the system has infinitely many solutions. Let's have a look at the corresponding coefficient matrix.
y - x = 1 3y - 3x = 3 → 1 & - 1 3 & - 3
We can calculate its determinant.
|A| = |
cc
a & b
c & d
|
=
ad-bc
2.3cm|
cc
1 & - 1
3 & - 3
| = (1)(-3)-(- 1)(3)
1.5cm= 0
As the determinant is zero, we cannot solve it using matrices. However, this fact is not enough to let us know if it has infinite solutions or no solution at all. When solving this algebraically we can know the difference. For a system with infinitely many solutions we get an identity, and for a no-solution system we obtain a false statement.
