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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} $?

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  • $\begingroup$ Are you expecting the equation to be true for $a, b,c$ given or for all $a,b ,c$? $\endgroup$ – mathcounterexamples.net Jun 5 at 14:27
  • $\begingroup$ @mathcounterexamples.net: I hope so. For any a,b,c. $\endgroup$ – Money Oriented Programmer Jun 5 at 14:28
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You can't; that matrix has no inverse.

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$$X=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$$

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I don't know how to put it, but it's basically a permutation matrix: $$X=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}$$

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$X$ 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

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