Prove that each matrix is invertible and find their inverses

Question

I have three $n\times n $ matrices $A,B,C$ and it is given that $ABC=I$, I should prove that each $A,B$ and $C$ are invertible and find their inverse.

Here is what I have: Since $ABC=I$ then I might have that $A$ and $B$ and $C$ are inverses of each other, but from this moment I got confused.

Answer 1

$A$ is invertible because $BC$ is its inverse.

$C$ is invertible because $AB$ is its inverse.

$B=A^{-1}IC^{-1}$, a product of invertible matrices, and is therefore invertible. To find $B^{-1}$, we have $$B^{-1}=(A^{-1}C^{-1})^{-1}=CA$$

Answer 2

Associativity gives $$I=A(BC)=(AB)C$$these should yield $A^{-1}$ and $C^{-1}$ pretty easily. What do you think we can do for $B^{-1}$ (hint: the previous part says $A$ and $C$ are invertible)?