Solving a linear homogeneous system setting the determinant equal to zero - theorem?

Question

I often face a matrix problem of this form: AB=CB. To solve it they write (A-C)B=0 and then to find the solutions det(A-C)=0. So is there a theorem about linear systems that tells us how to find the solution in this way? If I imagine a 2X2 matrix I would have an homogeneous system like this:

(a b)(x)=0

(c f)(y)=0

so why should I find the det = 0? I will just get ad - bc = 0 and from there I can find x and y?

Answer 1

I guess A and C depend on some variable here?

If $\det(A-C) \neq 0$ then $A - C$ is invertible, so the only solution is

$$B = (A-C)^{-1}0 = 0.$$

Conversely if $\det(A-C) = 0$ then $A-C$ has a nontrivial kernel so there's some nonzero solution for B.