Find an orthogonal matrix $P$ such that $PAP^{-1}=B$

Question

Find an orthogonal matrix $P$ such that $PAP^{-1}=B$, where

$$A = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix}$$

I know that:

• $A$ can be transformed to $B$ by first switching the first two rows, then the first two columns,

• permutation matrices are orthogonal.

But I am unable to proceed further in order to find matrix $P$. Please help me to solve this question.

Answer 1

What permutation matrix $P$ has the effect of swapping the first two rows of a matrix $A$ in the product $PA$?

This, and what you have written, immediately leads to the answer: $$P=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$$