Showing A^-1 has eigenvalues reciprocal to A's eigenvalues

[unfunf22]

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$$\lambda$$ - A) = 0
B = C^-1AC (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$$\lambda$$ - 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^-1AC

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

-Ian

 

[HallsofIvy]

Are you required to use "similar matrices"? If $\lambda$ is an eigenvalue of A, with eigenvector v, then $Av= \lambda v$. Now take $A^{-1}$ of both sides of that equation.

 

[unfunf22]

Well, if I didn't then I wouldn't be learning anything about them. I mean, most of these proofs I don't see how to use similar matricies to figure them out... There's a huge number:

a) If A can be diagonalized, then its eigenvalues are distinct. I said this is false. I just used a counter example in the form of the 3x3 identity matrix. How would I use similar matricies for this??

b) If all eigenvalues of A are equal to 2, then B^-1AB = 2I for some nonsingular B. I see how this one relates to similar matricies, but have no clue on how to prove it.

c) if all eigenvalues of A are zero, then A is the zero matrix. (No clue how to prove)

d) If all eigenvalues of A are zero, then A is similar to the zero matrix.

e) If B = C^-1AC then B³ = C^-1A³C (I think I can prove this using determinants)

f) If A is nonsingular, then all its eigenvalues are nonzero. - This is true, but it easily follows from my proof I presented earlier. I did not use similar matricies.

g) There exist 3x3 real matricies with no real eigenvalue (I don't see how similar matrices comes into play here)


I have a test today, so that is why I am trying to figure this stuff out. Any idea on how similar values would apply to any of these proofs? Any idea how I would prove some of them?
-Ian