Find an invertible matrix P and a diagonal matrix D so that (P^−1)AP = D, where
I'll start you off.
Find your charpoly:
The roots of this quadratic are your eigenvalues.
They are 1 and 6.
Sub them in for lambda in the matrix we derived, getting:
Use rref on these matrices and we find:
Your solutions are:
Bases for the eigenspace are:
Let this be P:
Now, find . You have all the necessary info.
Check me out. Easy to flub up.
I hope this helps.
If you don't have one, get a good linear algebra text. Anton and Rorres publish some good ones.