# 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.