Showing a set of matrices is a subspace I have to show that a set of matrices that commute with the matrix
$$\begin{bmatrix}
 0&a_2  &a_3  &\cdots   & a_n\\ 
 0& 0 &0  &\cdots  &0 \\ 
 \vdots &\vdots  &\vdots  &\ddots   &\vdots \\ 
 0&0  &0  &\cdots  &0 
\end{bmatrix}$$
 is a subspace and I have to find its basis.
I know that for proving a subspace I have to show that the zero matrix is in this set and that if $A$ and $B$ is in the set than also $A+B$ is in the set and also for if $r \in \mathbb{R}$ and matrix A is from the set than also $rA$ is from this set.
I also understand that two matrices commute if $AB=BA$ but I don't know with which matrix should I multiply the given one.
Ok, so I took an arbitrary matrix from $\mathbb{R}^{n \times n}$
$$
\begin{bmatrix}
e_{11} &e_{12} &e_{13} & \cdots & e_{1n} \\
e_{21} & \cdots&\cdots&\cdots&e_{2n} \\
\vdots & \vdots &\vdots &\ddots & \vdots \\
e_{n1} & \cdots & \cdots & \cdots &e_{nn}
\end{bmatrix}
$$
If I try to multiply it with the given one I get


*

*$$\begin{bmatrix}
e_{11} &e_{12} &e_{13} & \cdots & e_{1n} \\
e_{21} & \cdots&\cdots&\cdots&e_{2n} \\
\vdots & \vdots &\vdots &\ddots & \vdots \\
e_{n1} & \cdots & \cdots & \cdots &e_{nn}
\end{bmatrix}
\begin{bmatrix}
 0&a_2  &a_3  &\cdots   & a_n\\ 
 0& 0 &0  &\cdots  &0 \\ 
 \vdots &\vdots  &\vdots  &\ddots   &\vdots \\ 
 0&0  &0  &\cdots  &0 
\end{bmatrix}=
\begin{bmatrix}
 0&e_{11}a_2  &e_{11}a_3  &\cdots   &e_{11} a_n\\ 
 0& 0 &0  &\cdots  &0 \\ 
 \vdots &\vdots  &\vdots  &\ddots   &\vdots \\ 
 0&0  &0  &\cdots  &0 
\end{bmatrix}
$$


2.
$$
\begin{bmatrix}
 0&a_2  &a_3  &\cdots   & a_n\\ 
 0& 0 &0  &\cdots  &0 \\ 
 \vdots &\vdots  &\vdots  &\ddots   &\vdots \\ 
 0&0  &0  &\cdots  &0 
\end{bmatrix}
\begin{bmatrix}
e_{11} &e_{12} &e_{13} & \cdots & e_{1n} \\
e_{21} & \cdots&\cdots&\cdots&e_{2n} \\
\vdots & \vdots &\vdots &\ddots & \vdots \\
e_{n1} & \cdots & \cdots & \cdots &e_{nn}
\end{bmatrix}
 =
\begin{bmatrix}
a_2 e_{21}+a_3 e_{31}+ \cdots + a_n e_{n1} & a_2 e_{22}+\cdots+a_n e_{n2} & \cdots & \cdots & a_2 e_{2n}+ \cdots + a_n e_{nn} \\
0&0&\cdots&\cdots&0 \\
\vdots & \vdots & \vdots & \ddots &0 \\
0 &0&0 & \cdots &0
\end{bmatrix}$$
Now I have to compare the elements in the both computed matrices and I get a system of equations which I don't know how to solve.

Hint: The map $T:\Bbb R^{n \times n} \to \Bbb R^{n \times n}$ given by
$$
T(B) = AB - BA
$$
is a linear transformation.  As such, the kernel of $T$ is a subspace of $\Bbb R^{n \times n}$.
