Show that A has an inverse if all eigenvalues have negative real parts

[Physics_wiz]

Assume that all eigenvalues of an nxn real matrix A have negative real parts. Show that the inverse of A exists.

I really have no idea how to even start this one. Any hints?

 

[Dick]

If the inverse of A doesn't exist, then there is an x such that Ax=0. Doesn't this mean it has an eigenvalue that DOESN'T have a negative real part?