# Relationship between eigenvalues $A$ and $A^{-1}$ proof [closed]

4.15 Assume that $A$ is a nonsingular $n × n$ matrix.

(a) What is the relationship between the eigenvalues of $A$ and those of $A^{-1}$? Prove your answer.

(b) What is the relationship between the eigenvectors of $A$ and those of $A^{-1}$? Prove your answer.

## closed as off-topic by user296602, Misha Lavrov, Lord Shark the Unknown, A. Goodier, Xander HendersonMar 14 '18 at 13:44

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Community, Misha Lavrov, Lord Shark the Unknown, A. Goodier, Xander Henderson
If this question can be reworded to fit the rules in the help center, please edit the question.

• Welcome to MSE! It seems you've misunderstood the purpose of the site with your last two questions: this is a site to get assistance, but it is not a do-my-homework site. Please edit accordingly. – user296602 Mar 14 '18 at 0:27
• As it has been said before, welcome to the site! And yes, it is the etiquette to show some effort in solving the problem. That being said, let me just share that the eigenvalues of the inverse of a square invertible matrix are simply the inverse of the original eigenvalues. The vectors stay the same. – Antoni Parellada Mar 14 '18 at 0:47

Hint: if $Av=\lambda v$, then multiply both sides by $\frac{1}{\lambda} A^{-1}$.