This question is broken up into 4 parts to help you get to the right answer. Below is where I am so far.

Find eigenvalues of A:

A=[2 13

-1 -12]

det(lI - A) = 0

So the equation is l^2-10l - 11=0

So lambda = -11,1

b) Find corresponding eigenvectors

if l=-11

eigenvector is [-1

1]

if l=1

Eigenvector is [-13

1]

c) Find a matrix P that diagonalizes A

P = [-1 -13

1 1]

d) Verify that P^-1 A P is diagonal

This verification does not work for me, so I assume I did something wrong above? Any idea where I went wrong?