A surjectivity problem for 3 by 3 matrices

We begin this section by presenting the powers of the matrix A in a suitable form so the technicalities in the main section are minimal. In addition it enables us to obtain immediately a spectral mapping theorem. There are three cases to consider, namely (x− y)(x− z)(y− z) 6= 0, x = y = z , and finally PA(λ ) = (λ − x) (λ − y) with x 6= y . 1. Suppose (x− y)(x− z)(y− z) 6= 0. Then PA(λ ) = (λ − x)(λ − y)(λ − z) and from the Hamilton-Cayley Theorem we have PA(A) = (A−xI3)(A−yI3)(A−zI3) = 03; the null matrix of order three.