linear algebra - Similar matrices proof

Good evening, I'm stuck on how to proceed in the following qiestion.

Let A and B be nxn matrices and show that if there is a 𝜆 ∈ ℝ such that 𝐴−𝜆𝐼 is similar to 𝐵−𝜆𝐼, then 𝐴 is similar to 𝐵.

I thought to use determinant for both sides, but I'm not sure if it's the right way.

Thanks in advance!

Answer

Suppose $A-\lambda I$ and $B-\lambda I$ are similar.
By definition, we can find $S$ such that
$$A-\lambda I=S^{-1}(B-\lambda I)S = S^{-1} B S - \lambda I$$
Adding $\lambda I$ to the leftmost and rightmost sides of this equality reveals that $A$ and $B$ are also similar.