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

• unfunf22
In summary, Ian showed that if A is nonsingular, then A-1 is undefined, det(A-1) = 1/det(A), and if lambda is not 0, then det(I\lambda - A) = 0. He is lost on how to use the idea of similar matrices in his proof.
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

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.

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

## 1. What is meant by "Showing A^-1 has eigenvalues reciprocal to A's eigenvalues"?

This means that if A has eigenvalues λ1, λ2, ..., λn, then the eigenvalues of A^-1 will be 1/λ1, 1/λ2, ..., 1/λn.

## 2. Why is it important to show that A^-1 has eigenvalues reciprocal to A's eigenvalues?

This is important because it helps us understand the relationship between a matrix and its inverse. It also allows us to find the eigenvalues of A^-1 without directly calculating the inverse.

## 3. How can we prove that A^-1 has eigenvalues reciprocal to A's eigenvalues?

One way to prove this is by using the fact that the eigenvalues of a matrix are the roots of its characteristic polynomial. We can then show that the characteristic polynomial of A^-1 is the reciprocal of the characteristic polynomial of A.

## 4. Can this property hold true for all matrices?

No, this property only holds true for invertible matrices. If a matrix is not invertible, then its inverse does not exist and this property cannot be applied.

## 5. How can this property be useful in real-world applications?

This property can be useful in solving systems of linear equations and in finding the eigenvalues and eigenvectors of a given matrix. It can also help in understanding the relationship between a system and its inverse, which can have practical applications in fields such as engineering and physics.

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