Theorem:

I've a question regarding the proof of a theorem.

There is a theorem in my linear algebra book that states:

"If and are similar x matrices, then they have the same eigenvalues."

Proof of this theorem:

Let and be similar matrices so there exist an invertible matrix such that

By the properties of determinant it follows that:

My question:

My question is: why is that

How do you proof that left hand side of this statement is equal to the right hand side?

What property of linear algebra makes this true? I can't figure out the answer for this problem. Is it possible help me finding the answer?