Determine the characteristic polynomial for a

Question

Questions:

1) This question concerns the linear transformation ƒ : i 3 → i 3 represented by the matrix

                                   A =  (-1  -1    1)
                                          ( 2   -1    2)   
                                          ( 2    1    0)

With respect to the standard basis in both the domain and codomain.

(a) Determine the characteristic polynomial for A.

(b) Show that λ = 1 is an eigenvalue and find the remaining eigenvalues of A.

(c) Determine all the corresponding eigenvectors and eigenspaces.

(d) Write down a matrix P and a diagonal matrix D such that P -1 AP = D

(e) Hence verify that P -1 AP = D by matrix multiplication.