$P$ is an matrix invertible Proof $|\lambda I-PBP^{-1}|=|\lambda I -B|$

Question

I have this problem :

$P$ is an matrix invertible

Proof : $|\lambda I-PBP^{-1}|=|\lambda I -B|$

I'm not so sure about my answer, since I don't think I could use "double" determinant for example $||B||$ instand of $|B|$.

This is what I did (notice the notation):

$|\lambda I-PBP^{-1}|=|\lambda I-|P||B||P^{-1}|| = |\lambda I-|I||B|| =|\lambda I-B|$

I pretty sure I cannot use determinant that way, and since $|\lambda I-PBP^{-1}|\neq|\lambda I| -|P||B||P^{-1}$|

I don't find any determinant property to use in order to solve this problem.

Any Ideas?

Thanks!

Answer 1

That's not quite right. You generally have the right idea though. Here's a hint:

$$|\lambda I-PBP^{-1}| = |P(\lambda I - B)P^{-1}|.$$