Find all matrices that commute with $A$ 
Given $$A = \begin{bmatrix}
3 & 1 &0 \\ 
0 &3  & 1\\ 
0 &0  & 3
\end{bmatrix}$$ find matrices $B$ such that $AB=BA$.

Trivially $B=A^{-1}$ and $B=kI$ are the solutions
Also we have Characteristic Polynomial as
$$A^3-9A^2+27A-27I=0$$ $\implies$
$$(A-3I)^3=0$$
Is it possible to find other $B's$ using above Nilpotency of $A-3I$?

By just writing out the matrix multiplication and simplifying you get:
\begin{align*}
AB &= BA\\
\begin{bmatrix}
3 & 1 &0 \\ 
0 &3  & 1\\ 
0 &0  & 3
\end{bmatrix}
\begin{bmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33} \\
\end{bmatrix}
&=
\begin{bmatrix}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33} \\
\end{bmatrix}
\begin{bmatrix}
3 & 1 &0 \\ 
0 &3  & 1\\ 
0 &0  & 3
\end{bmatrix}\\
\begin{bmatrix}
 3b_{11} + b_{21} & 3b_{12} + b_{22} & 3b_{13}+b_{23} \\
 3b_{21} + b_{31} & 3b_{22} + b_{32} & 3b_{23}+b_{33} \\
 3b_{31} & 3b_{32} & 3b_{33}
\end{bmatrix}
&=
\begin{bmatrix}
3b_{11} & b_{11}+3b_{12} & b_{12} + 3b_{13} \\
3b_{21} & b_{21}+3b_{22} & b_{22} + 3b_{23} \\
3b_{31} & b_{31}+3b_{32} & b_{32} + 3b_{33}
\end{bmatrix} \\
\begin{bmatrix}
 b_{21} & b_{22} & b_{23} \\
 b_{31} & b_{32} & b_{33} \\
 0 & 0 & 0
\end{bmatrix}
&=
\begin{bmatrix}
 0 & b_{11} & b_{12} \\
 0 & b_{21} & b_{22}  \\
 0 & b_{31} & b_{32}
\end{bmatrix}
\end{align*}
Hence, $b_{21},b_{31},b_{32}=0$, $b_{11}=b_{22}=b_{33}$ and $b_{12}=b_{23}$, confirming the solutions are exactly those given by Robert.
