# Thread: eigenvalues of A inverse

1. ## eigenvalues of A inverse

I need to prove the following: If matrix A is nonsingular, then the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.

Here is what I know:

det(A-lambdaI)=0

det(A inverse)=1/det(A)

I feel like I need to use this second fact somehow in my proof, but I am struggling with how to get to this step.

Help!

2. Originally Posted by ecc5
I need to prove the following: If matrix A is nonsingular, then the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.

Here is what I know:

det(A-lambdaI)=0

det(A inverse)=1/det(A)

I feel like I need to use this second fact somehow in my proof, but I am struggling with how to get to this step.

Help!

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

Tonio

3. Oh!! Thank you! I was definitely overthinking it.

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### if A is non-singular matrix prove that the eigenvalues of A-1 are the reciprocal of eigen values of A.

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