- #1

EvLer

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Notation: Q' - inverse of Q, != stands for "not equal";

Suppose A and B are nxn matrices such that A = QBQ' for some invertible matrix Q. Prove that A and B have the same characteristic polynomials

I can prove that they have the same determinant, but that is about it. I know that charact. polyn. looks like so:

det(A - I*lambda) = det(B - I*lambda)

det(QBQ' - I*lamda) = det(B - I*lambda)

It would be equal if Q and Q' cancel out, but isn't it true that det(A + B) != det(A) + det(B).

I am not sure where to go from here. Is it correct to multiply expressions inside parenthesis by something on both sides? even if it's in determinant

I am studying for the final, so any help is appreciated more than ever.