I have the following HW problem:
Suppose $A$ and $B$ are $n\times n$ matrices with $AC=CB$ for an invertible matrix $C$. Also let $B-\lambda I_n$ be non-invertible for a real number $\lambda$. Prove that $A-\lambda I_n$ is also non-invertible.
I know that $\det{A} = 0 \iff$ the system $A\vec{x} = \vec{0}$ has a non-zero solution. So I let $(A-\lambda I_n)\vec{x}=\vec{0}$ and then rewrote this as $A\vec{x} = \lambda\vec{x}$. But I'm unsure what properties I can use from this point on to actually prove that this is the case (and also show $\vec{x} \ne \vec{0}$; that is, that the system has a non-zero solution).
Thank you!
Edit: Didn't cover similar matrices, characteristic polynomials, or eigenvalues yet