Question

Find an invertible matrix p such that p−1ap is diagonal

Answer 1

The PP matrix is just the matrix of eigenvectors of A. You find the eigenvectors by solving the equation

(λI−A)x=0(λI−A)x=0

For eigenvalue λ=0λ=0, you get the associated eigenvector ⎡⎣⎢1−31⎤⎦⎥[1−31]

For eigenvalue λ=−3,λ=−3, you get the associated eigenvector ⎡⎣⎢101⎤⎦⎥[101]

Fr eigenvalue λ=1,λ=1, you get the associated eigenvector ⎡⎣⎢.5−11⎤⎦⎥[.5−11]

Thus your matrix P=⎡⎣⎢1−31101.5−11⎤⎦