# Finding $X$ from $X \left(\begin{smallmatrix} a \\ b \\c \end{smallmatrix}\right) = \left(\begin{smallmatrix} b \\ c \\a \end{smallmatrix}\right)$

I am trying to solve the following matrix equation.

$$X \begin{pmatrix} a \\ b \\c \end{pmatrix} = \begin{pmatrix} b \\ c \\a \end{pmatrix}$$

My analysis:

• $$X$$ must be 3 by 3 matrix.
• $$X=\begin{pmatrix} b \\ c \\a \end{pmatrix} \begin{pmatrix} a \\ b \\c \end{pmatrix}^{-1}$$

# Question

How can I calculate $$\begin{pmatrix} a \\ b \\c \end{pmatrix}^{-1}$$?

• Are you expecting the equation to be true for $a, b,c$ given or for all $a,b ,c$? – mathcounterexamples.net Jun 5 '19 at 14:27
• @mathcounterexamples.net: I hope so. For any a,b,c. – Money Oriented Programmer Jun 5 '19 at 14:28

On the other hand, you know that$$X.\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}$$and therefore the first column of $$X$$ will be $$\left[\begin{smallmatrix}0\\0\\1\end{smallmatrix}\right]$$. You can compute the other columns by the same method. You will get that$$X=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$$
I don't know how to put it, but it's basically a permutation matrix: $$X=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}$$
$$X$$ belongs to a special family of matrices called permutation matrices, which swap elements of the input. For more information, see here: https://en.wikipedia.org/wiki/Permutation_matrix