With a 2x2, you don't have to use the det method.
The characteristic polynomial is simply
Your eigenvalues are correct. Looks correct to me though.
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:
det(lI - A) = 0
So the equation is l^2-10l - 11=0
So lambda = -11,1
b) Find corresponding eigenvectors
eigenvector is [-1
Eigenvector is [-13
c) Find a matrix P that diagonalizes A
P = [-1 -13
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?