commuting matrices & polynomials 1

Question

I need help on this problem:

Problem:

Find two 3x3 matrices, A and B that commute with each other; and neither A is a polynomials of B nor B is a polynomial of A

Answer 1

Try $A = E_{13}$, the matrix with zeros everywhere except at (1,3), and $B=E_{22}$. Then $AB=0=BA$, and you can show that any polynomial in $A$ is equal to $\alpha A + \beta I$ for some scalars $\alpha $ and $\beta$, same for $B$. That makes it easy to see neither is a polynomial in the other.

Answer 2

A=diag(1,1,2) en B is the matrix with rows [1,1,0\0,1,0\0,0,1]. Then AB=BA, B is not polynomial in A (B is not a diagonal matrix) en A is not polynomial in B. For any polynomial p of degree <3 with P(B)=A should have the property p(1)=1 (since p([1,1\0,1])=diag(1,1), so p(x)=1+(x-1)^2) and p(1)=2.

J. Vermeer