How do I show that the eigenvalues of two square matrices of different dimensions are the same?

I have three matrices in a field $F$:

$X in F^{a,a}, Y in F^{b,b}, Z in F^{a,b}$, where $a,b in mathbb{N}, a geq b$ and $text{rank}(Z) = b$. The following term describes their relation:

$A cdot C = C cdot B$

I want to show that they have the same eigenvalues. I have started with the following about their determinant:

$text{det}(AC) = text{det}(CB) iff text{det}(A) = text{det}(B)$

However, I am not even sure if this is even relevant and how to go on from there.