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

Answer:

PROOF:

Suppose that A is an invertible non singular matrix.

Then,

$Ax=\lambda x\\\\A^{-1}Ax=A^{-1}\lambda x\\\\\frac{1}{\lambda}x=A^{-1}x\\\\\lambda^{-1}x=A^{-1}x$

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.