- #1

unfunf22

- 15

- 0

## Homework Statement

If A is nonsingular, prove that the eigenvalues of A

^{-1}are the reciprocals of the eigenvalues of A.

*Use the idea of similar matrices to prove this.

## Homework Equations

det(I[tex]\lambda[/tex] - A) = 0

B = C

^{-1}AC (B and A are similar, and thus have the same determinants)

## The Attempt at a Solution

At first I showed that A is nonsingular iff 0 is not an eigenvalue of A. To do this I just used the fact that det(A

^{-1}) = 1/det(A) and that if lambda was 0, then we'd have det(A) = 0, which would mean A

^{-1}is undefined. If lambda isn't 0, then we have det(I[tex]\lambda[/tex] - A) = 0, which tells us A is nonsingular.

As for the other proof, I'm convinced I have to use the idea of similar matrices, because the book I am using is focusing on them right now, and these exercises are relating to them.

But A

^{-1}and A are not the same linear transformation (unless A = I), so they are not similar. Therefore I cannot use the formula: B = C

^{-1}AC

So, I am lost on how to do this proof using the idea of similar matrices. Anyone know how I could accomplish this?

-Ian