Why does $A$ times its inverse equal to the identity matrix? I was trying to come up with a proof of why: $AA^{-1} = I$.
If we know that: $A^{-1}A = I$, then $A(A^{-1}A) = A \implies (AA^{-1})A = A$.
However I don't like setting $AA^{-1} = I$ for fear that it might be something else at this point, even though we know that $IA=A$. For example, could $A$ times its inverse equal something other than the identity leading back to the original matrix $A$.
Does anyone have a another proof for why $A$ times its inverse would give you the identity or could explain something I'm missing?

We say a matrix $B$ is an inverse for $A$ if $AB = BA = I$, and the notation for $B$ is $A^{-1}$.
So it's by definition $AA^{-1}=I$, you cannot really prove it.
