Show that $A$ can be inverted when $AB$ = Identity matrix

Question

If A and B are both n x n matrices and AB = I, how would you prove that A can be inverted?

Answer 1

$AB = I \implies det(AB) = det(I) \implies det(A)det(B) = 1 \implies det(A) \neq 0$. Therefore, $A$ is invertible.