# Thread: Simple matrix algebra identity

1. ## Simple matrix algebra identity

Hello everyone!

I'm stuck on this particular concept...
$AB=AC \ \Rightarrow \ A(B-C)=0.$ Soooo when is it that $B-C=0$ i.e. $B=C \ ... \ (1)$ but probably if $A^{-1}$ exists then we can multiply to the right by $A^{-1}$ so we get $(1)$. But how can we know before hand?
Thanks!

2. If you know that $\det(A)\not=0$, then $A$ is invertible, which would imply that $B=C.$ That's one condition that is sufficient. It's not necessary. As a counterexample, just let $A=0$, and $B=C=I.$

Note: you're actually going to want to multiply on the left by $A^{-1}$, if it exists, in order to eliminate the $A$'s.