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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} $ and $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. Thanks.

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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: $$P=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$$

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  • $\begingroup$ Why are you not taking $P^{-1}$ into account? Just $PA$ suffices? $\endgroup$
    – s1mple
    Apr 4, 2020 at 7:14
  • $\begingroup$ And how did you directly get $P$? Trial and error? $\endgroup$
    – s1mple
    Apr 4, 2020 at 7:15
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    $\begingroup$ @s1mple Since permutation matrices are orthogonal, $P^{-1}=P^T$. It turns out that $P$ is its own transpose, and hence $P$ is its own inverse, and hence $AP$ swaps the first two columns of $A$ as you wrote. $\endgroup$ Apr 4, 2020 at 7:15
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    $\begingroup$ @s1mple I got $P$ from your description. $\endgroup$ Apr 4, 2020 at 7:16

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