Let A be an n × n orthogonal matrix. Either prove or disprove that any matrix similar to A is also orthogonal.

Question

This is the question I have, I think the statement is false but am at a loss at how to proceed disproving it. I am struggling to come up with a nice counter example and have no idea how else to approach this.

I tried to write it as

$P^{-1}QDQ^{-1}P=B$ and try and go about it that way but got stuck.

Answer 1

Hint: The matrix $$ A = \pmatrix{1 & 0\\0 & -1} $$ is orthogonal. Which matrices are similar to $A$? Can you come up with an example that is not orthogonal?