Non-singularity of a matrix

Question

Let A be any complex matrix, prove that $\ I+A^* A $ is non-singular.

I just need some hint in how to proceed with the question.

Answer 1

If the matrix is singular then there exists a vector such that,

$$ (I + A^* A) v = 0$$

which implies that $v$ is an eigenvector of $A^*A$ with eigenvalue equal to $-1$.

You should be able to finish the proof from there.