# Thread: Proof for inverse matrices

1. ## Proof for inverse matrices

Prove that for three non-singular matrices A, B, C that $\displaystyle (ABC)^{-1}=C^{-1}B^{-1}A^{-1}$

I know that for two square matrices, C, D, say, $\displaystyle (CD)^{-1}=D^{-1}C^{-1}$. I tried using this result to prove $\displaystyle (ABC)^{-1}=C^{-1}B^{-1}A^{-1}$, by considering AB as one square matrix, and multiplying by C, and here is my problem.
$\displaystyle (AB)^{-1}=B^{-1}A^{-1}$, I don't know how to continue after this. Or am I off the right track?

Thanks!

2. You're on the right track. Since we know that for all nonsingular matrices $\displaystyle A,B$, that $\displaystyle (AB)^{-1}=B^{-1}A^{-1}$
So for three nonsingular matrices $\displaystyle (ABC)^{-1}=((AB)C)^{-1}=C^{-1}(AB)^{-1}=C^{-1}B^{-1}A^{-1}$

3. You may also verify the rule directly.

Recall that two matrices $\displaystyle A,B$ are inverses if $\displaystyle AB=BA=I$. We claim that $\displaystyle (ABC)^{-1}=C^{-1}B^{-1}A^{-1}$. We can check it by computing $\displaystyle ABC(C^{-1}B^{-1}A^{-1})$ and $\displaystyle (C^{-1}B^{-1}A^{-1})ABC$. Everything should work out nicely.

4. Originally Posted by MattMan
You're on the right track. Since we know that for all nonsingular matrices $\displaystyle A,B$, that $\displaystyle (AB)^{-1}=B^{-1}A^{-1}$
So for three nonsingular matrices $\displaystyle (ABC)^{-1}=((AB)C)^{-1}=C^{-1}(AB)^{-1}=C^{-1}B^{-1}A^{-1}$
thank you very much, I wasn't sure how exactly to work it out from where I left.