<s>[INST] Finding the inverse of a matrix given an equation So I've been given this equation:
$A\begin{bmatrix}
2&3&1&5\\
1&0&3&1\\
0&2&-3&2\\
0&2&3&1
\end{bmatrix} = \begin{bmatrix}
0&1&0&0\\
0&0&0&1\\
0&0&1&0\\
1&0&0&0
\end{bmatrix}$
I'm supposed to find the inverse of A using this but I'm not really sure where to start. Any ideas?
 [/INST] If $AB=C$, then $A^{-1}=BC^{-1}$.
Now if we interpret the right-hand side matrix as a change of basis matrix, such that
$$u_1=e_4,\enspace u_2=e_1,\enspace u_3=e_3, \enspace u_4=e_2,$$
then, conversely
$$ e_1=u_2,\enspace e_2=u_4,\enspace e_3=u_3,\enspace e_4=u_1,$$
so that
$$\begin{bmatrix}
0&1&0&0\\
0&0&0&1\\
0&0&1&0\\
1&0&0&0
\end{bmatrix}^{-1}=\begin{bmatrix}
0&0&0&1\\
1&0&0&0\\
0&0&1&0\\
0&1&0&0
\end{bmatrix}$$
and finally
$$BC^{-1}=\begin{bmatrix}
0&2&3&1\\
2&3&1&5\\
0&2&-3&2\\
1&0&3&1
\end{bmatrix}.$$
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