Linear algebra. invertible matrix and its eigenvalue

[mcbonov]

Homework Statement

Let A be an invertible matrix.
show that if λ is an eigenvalue of A,
then 1/λ is an eigenvalue of A^-1


PLEASE HELP ME .
Thank you.

 

[Mark44]

Homework Statement

Let A be an invertible matrix.
show that if λ is an eigenvalue of A,
then 1/λ is an eigenvalue of A^-1


PLEASE HELP ME .
Thank you.

You're new here (welcome to Physics Forums!) so you probably haven't had a chance to check out the rules. Before we can provide any help, you need to show an effort at the problem you're posting.

When you're working proofs such as these, it's crucial that you have the definitions of the terms being used.

For example, what does it mean that A is an invertible matrix?
What does it mean that λ is an eigenvalue of A?
What's another way of saying that 1/λ is an eigenvalue of A^-1?

 

[mcbonov]

i tried Ax =λx
(Ax-λx)=0
(A-λ)x=0 since x is not a zero vector
A-λ=0 then A=λ

THEN A^-1 = λ^-1
so (A^-1)x = (λ^-1)x

I basically don't know how to prove...
I don't think the above is near to correct answer.
that's the most I can do ...
how can i solve this problem??

 

[Mark44]

i tried Ax =λx
(Ax-λx)=0
(A-λ)x=0 since x is not a zero vector

No, you can't say this. A is a matrix, but λ is a scalar. You can't subtract a scalar from a matrix. What you can say is this:
(A-λI)x=0, for some nonzero vector x (an eigenvector of the eigenvalue λ).

A-λ=0 then A=λ

Makes no sense, since A and λ are two completely different kinds of things.

THEN A^-1 = λ^-1
so (A^-1)x = (λ^-1)x

I basically don't know how to prove...
I don't think the above is near to correct answer.
that's the most I can do ...
how can i solve this problem??

You know that Ax =λx for some nonzero vector x. You also know that A is invertible. What happens if you multiply both sides of the equation, on the left, by A^-1.

Keep in mind what you would like to end up with, that would imply that 1/λ is an eigenvalue of A^-1.