Why does $A$ times its inverse equal to the identity matrix? [duplicate]

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?

marked as duplicate by almagest, hardmath, user223391, Oscar Cunningham, CrostulJun 16 '16 at 14:15

• How do you define $A^{-1}$ if not that it is $AA^{-1}=I$? – Hirshy Jun 16 '16 at 13:54
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.
• The question is asking why $AB=I$ implies $BA=I$. The use of the word "inverse" or the symbol $A^{-1}$ is a red herring. – almagest Jun 16 '16 at 14:15