Is 1/λ an Eigenvalue of the Inverse of A?

[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?

 

[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

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

?