Question

True or false. Linear Algebra.
If A is invertible and 1 is an eigenvalue for A, then 1 is also an eigenvalue of A^-1.

Answer 1

If 1 is an eigenvalue for the matrix A, then

Ax = 1*x

Ax = x

A^(-1)*Ax = A^(-1)x

x = A^(-1)x

A^(-1)x = x

A^(-1)x = 1*x

So 1 is also an eigenvalue of A^(-1)

Answer 2

nice use of definition