origin415 wrote:If the determinant is nonzero than there exists exactly one solution. If the determinant is zero, there could be no solutions, or there could be infinitely many. It just means the matrix isn't invertible.
As a trivial example, the system of equations
x + 0y = 0
0x + 0y = 0
has a coefficient matrix of
1 0
0 0
Which has determinant zero. However, there are infinite solutions, x = 0 and y is anything.
Changing this, you could have
x + 0y = 0
0x + 0y = 1
Which is obviously absurd, so has no solutions.
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