Theorem EIM says that if λ is an eigenvalue of the nonsingular matrix A, then 1 is an eigenvalue of A−1

Since λ is an eigenvalue of a nonsingular matrix, λ ≠ 0 (Theorem SMZE). 

A is invertible (Theorem NI), and so −λA is invertible (Theorem MISM).

Thus −λA is nonsingular (Theorem NI) and det (−λA) ≠0 (Theorem SMZD).

So 1/ λis a root of the characteristic polynomial of A⁻¹ and so is an eigenvalue of A⁻¹. 

This proof is due to Sara Bucht.