I wonder how to construct such Matrix that in such form: $$A=\begin{bmatrix} f(x) & g(x)\\ 0& h(x)\\ \end{bmatrix}$$ Satisfying $f,g,h$ are not homogeneous,
case 1: $A^2\neq\begin{bmatrix}d_1(x) & 0\\ 0 & d_2(x)\\\end{bmatrix}$, but $A^3=\begin{bmatrix}d_1(x) & 0\\ 0 & d_2(x)\\\end{bmatrix}$ , for some $d_1(x),d_2(x)$
case 2: $A^kA'=A'A^k,k\geq3$,$A^2A'\neq A'A^2$
Case 1: $A^2 = \left [\begin{matrix} f^2 & g(f+h)\\0 & h^2\end{matrix}\right]$, and $A^3 = \left [\begin{matrix} f^3 & g(f^2 + fh +h^2)\\0 & h^3\end{matrix}\right]$. It doesn't take much to get your answer from there.