Suppose that B is the inverse of A. Show that if |psi> is an eigenvector of A with eigenvalue a not equal to 0, then |psi> is an eigenvector of B with eigenvalue 1/a.(adsbygoogle = window.adsbygoogle || []).push({});

So I know that A|psi> = a|psi>, and I'm trying to prove that A^(-1)|psi> = 1/a|psi>. I tried simplifying A as a 2x2 matrix and then doing the inverse of that. And then I assumed that the inverse of A has an eigenvalue b. So then I did the determinant of A^(-1)-b = 0 in the hopes to find b and see that it's equal to 1/a. But that became really messy.

Any suggestions on how to solve this problem? Thank you so much!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Eigenvalues and Eigenvectors

**Physics Forums | Science Articles, Homework Help, Discussion**