# 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!

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Your computation is wrong. You cannot just replace $AC$ with $(AC)^{-1}$. – Martin Argerami Sep 15 '13 at 1:00
I don't see any instances of that in my question – codedude Sep 15 '13 at 1:01
oh nevermind now I do. – codedude Sep 15 '13 at 1:02
It is exactly what you did in your second equality: you wrote $ACA=(C^{-1}A^{-1})A$. Not true with the exception of some very particular choices of $A$ and $C$. – Martin Argerami Sep 15 '13 at 1:02
If you use bmatrix instead of matrix, you get the square brackets. – Brian M. Scott Sep 15 '13 at 1:15

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.
You have $$C=A^{-1}(ACA)A^{-1}.$$ So $$C= \begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 4 & 1 & 2 \\ \end{bmatrix}\,\cdot\, \begin{bmatrix} 2 & 1 & 3 \\ -1 & 2 & 2 \\ 2 & 1 & 4 \\ \end{bmatrix}\,\cdot\, \begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 4 & 1 & 2 \\ \end{bmatrix}.$$ Now you can just perform the computation.