# How do you find the inverse of A=((1, 0, 0, 0), (0, 0, 1, 0), (0,0,0, 1), (0, 1, 0, 0))?

Jul 16, 2016

$\left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{matrix}\right)$

#### Explanation:

$A = \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{matrix}\right)$

Notice what this matrix does as a linear transformation:

$\left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{matrix}\right) \left(\begin{matrix}a \\ b \\ c \\ d\end{matrix}\right) = \left(\begin{matrix}a \\ c \\ d \\ b\end{matrix}\right)$

It cyclically permutes three ordinates.

Therefore if applied $3$ times it results in the identity:

${A}^{3} = I$

So the inverse of $A$ is:

${A}^{2} = \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{matrix}\right) \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0\end{matrix}\right) = \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{matrix}\right)$