TT* + I is invertible I've the following exercise which I can't solve:
Prove that: 
$$ AA^* + I $$
is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $  A^* $ is the adjoint operator. 
Any help will be appreciated. 

Assuming that there is an inner product in $V$. Then for every $x \in V, x \neq 0$, we have 
$$x^*(AA^*+I)x = \underbrace{(A^*x)^*(A^*x)}_{\geq 0}+\underbrace{x^*x}_{>0}$$
So $AA^*+I$ is positive definite and thus invertible.
