Matrix Multiplication Problem I'm working on the following problem and I can't seem to come up with the right answer. 
$$ \text{Let}:  A^{-1} =
        \begin{bmatrix}
        1 & 2 & 1 \\
        0 & 3 & 1 \\
        4 & 1 & 2 \\
        \end{bmatrix}
$$
Find a matrix such that:
$$ ACA = 
\begin{bmatrix}
        2 & 1 & 3 \\
        -1 & 2 & 2 \\
        2 & 1 & 4 \\
        \end{bmatrix}
$$
Could someone point me in the right direction? Thanks!

Let $$ B = 
\begin{bmatrix}
        2 & 1 & 3 \\
        -1 & 2 & 2 \\
        2 & 1 & 4 \\
        \end{bmatrix}
$$
$ACA = B$ if and only if $A^{-1}ACA = A^{-1}B$ if and only if $A^{-1}ACAA^{-1} = A^{-1}BA^{-1}$. Now, multiplication between matrices is not commutative but it is associative! Hence you have:
$$A^{-1}ACAA^{-1} = (A^{-1}A)C(AA^{-1}) = C$$
Then to find such a matrix $C$ you just need to calculate $A^{-1}BA^{-1}$, which is something you can do explicitly.
