Question

Suppose that A is a nonsingular matrix with an eigenvalue λ. Show that 1/λ is then an eigenvalue of A-1 .

Answer 1

PROOF:

Suppose that A is an invertible non singular matrix.

Then,

`Ax=\lambda x\\\\A^(-1)Ax=A^(-1)\lambda x\\\\(1)/(\lambda)x=A^(-1)x\\\\\lambda^(-1)x=A^(-1)x`

This shows that if A is non singular invertible matrix ad if λ is eigenvalue of A then λ⁻¹ is eigenvalue of A⁻¹.