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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

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up vote 4 down vote accepted

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.

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