# If Y is orthogonal, does it follow that $Y^{-1}XY$ is orthogonal?

I am struggling with what my professor has written in my notes, because he hasn't proven this anywhere. In part of a longer proof, this comes up:

"Since Y is an orthogonal matrix, so is $Y^{-1}XY$. It follows that $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ is an orthogonal matrix, having determinant 1, so it is a rotation in the plane W"

$$Y^{-1}XY=\begin{pmatrix} 1 & 0 & 0 \\ 0 & a & b \\ 0 & c & d \end{pmatrix}$$

Is it true in general that if Y is orthogonal, $Y^{-1}XY$ is orthogonal?

This is part of a proof of this theorem: If X is a 3x3 orthogonal matrix with det X = 1, then X is given by rotation around some line L through the origin about some angle.

The question in the title is obviously false by taking $X=0$, say. But note that here $X$ is also orthogonal, and the product of any two or more orthogonal matrices is orthogonal (you should try proving this, it's quite simple). Since both $Y$ and $Y^{-1}$ are orthogonal, it should be clear that so is $Y^{-1}XY$.