Matrix question. ACA-AB=0. What's C?

[ABCD1234pop]

I know how to find A inverse. And I know ACA=AB.

But how do I go through solving this? I figured it had something to do with the inverse of A. But matrix multiplication isn't the same as normal multiplication.

Will something like

(Inverse of A)ACA=(Inverse of A)B

Since (Inverse of A)A=The identity Matrix

CA=B

Then

CA(Inverse of A)=B(Inverse of A)

C=B(Inverse of A)

I'm pretty sure all of this is wrong, but even if it isn't, I'm stuck here. How can I solve this and similar questions?

[timo]

Straightforward plugging into the equation will show you that your result is a correct solution (it doesn't show that it is the only solution, though):

[math]ACA - AB \stackrel{C=BA^{-1}}{=} AB\underbrace{A^{-1}A}_{=1} - AB = AB - AB = 0[/math]