# Math Help - Matrices 2

1. ## Matrices 2

Help! No matter how many times I do still can't get the right answer for C! Someone show the formula please?

A, B and C are square matrices such that $BA=B^{-1}$ and $ABC=(AB)^{-1}$. Show that $A^{-1}=B^2=C$. If $B=\left(\begin{array}{ccc}1&2&0\\0&-1&0\\1&0&1\end{array}\right)$, find C and A.

$A=\left(\begin{array}{ccc}1&0&0\\0&1&0\\-2&-2&1\end{array}\right)$ $C=\left(\begin{array}{ccc}1&0&0\\0&1&0\\2&2&1\end{ array}\right)$
Starting from $BA=B^{-1}$, multiply both sides by B on the left : $B^2A=BB^{-1}=I_2$ and then by $A^{-1}$ on the right : $B^2AA^{-1}=I_2A^{-1} \implies B^2=A^{-1}$
For the second equality, start from $ABC=(AB)^{-1} = B^{-1}A^{-1}$ and do the same