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. 

Assume that there's a $v\in V$ such that $AA^*v=-v$. Can you use the definition of the adjoint to conclude that $v=0$?
