Finding Eignevalues & Eigenvectors of A⁻¹ Without Direct Computation

[brad sue]

Hi, I need help for this problem:
Find the eignevalues and eingenvectors for the matrix below. DO NOT compute them directly by computing the matrix:
A^-1
We need to find some kind of demonstration to see if the eignevalues of A^-1 are the same, opposite or inverse (or whatever) as those of matrix A
Suppose that the eignvalues are 1,2,3 and the eignvectors are [1,1,0], [0,1,0],[ 3,-1,2] ( in columns
Thank you
B

 

[HallsofIvy]

If $$Ax= \lambda x$$ Then $$A^{-1}Ax= A^{-1}\lambda x= \lambda A^{-1}x$$.

What does that tell you?

 

[brad sue]

If $$Ax= \lambda x$$ Then $$A^{-1}Ax= A^{-1}\lambda x= \lambda A^{-1}x$$.
What does that tell you?

I have been thinking but I really do not know.
Can we say that I ( indentity matrix)= lambda*A^-1??
I does make me go far doesn't it?