Eigenvalue and invertible matrices

1. Dec 3, 2008

stine23

How do I prove that if A is an invertible matrix and lambda does not equal zero then one dived by lambda is an eigenvalue of the inverse of A?

2. Dec 3, 2008

Ben Niehoff

First of all, if A is invertible, then none of the eigenvalues can be 0, by definition.

Second, consider

$$Av = \lambda v$$

for some eigenvector v. What is

$$A^{-1}(\lambda v)$$

?