Set of matrices with some properties Let there be a set $$ \mathscr M⊆\mathbf M_3^\Bbb C,\mathscr M≠∅ $$
with the following properties
$$A,B∈ \mathscr M\Longrightarrow A+B∈ \mathscr M $$
$$ A∈\mathscr M,C∈M_3^\Bbb C\Longrightarrow CA∈\mathscr M $$
$$ X∈M_{3,1}^\Bbb C,AX=O,\forall A∈\mathscr M \Longrightarrow X=0.$$
Prove that
 $$ \mathscr M=\mathbf M_3^\Bbb C.$$
 A: Let $E_{i,j}$ be a matrix consisting of all zeroes except for a $1$ on the $i$-th row and the $j$-th column. This matrix has the property that for another matrix $M$ the product $E_{i,j}M$ has all zeroes except for the $i$-th row of $M$ that shows up at the $j$ -th row. From this we conclude that the matrices of $\mathscr M $ are of the form $\begin{pmatrix}u_1 & u_2 & u_3\\v_1 & v_2 & v_3\\ w_1 & w_2 & w_3 \end{pmatrix}$ where $(u_1,u_2,u_3), (v_1,v_2,v_3), (w_1,w_2,w_3)$ are rows in a subspace $W $ of $\mathbf M_{1,3}^\Bbb C$, the set of $1 \times 3$ matrices. If $W$ has dimension $3$ we're done. If not then there exists a non-zero column $X = \begin{pmatrix}z_1 \\ z_2 \\ z_3 \end{pmatrix}$ such that $AX = 0$  $ \forall A \in \mathscr M $, but this is in contradiction with the last property.
A: Let $M_3(\mathbb{C})$ be the set of $3 \times 3$ matrices over $\mathbb{C}$. Note that your first two conditions on $M$ together with $M \neq \emptyset$ are telling you that $M$ is a left ideal in $M_3(\mathbb{C})$. Such ideals can be described, see for example What are the left and right ideals of matrix ring? How about the two sided ideals?. Now combine this with your last condition on $M$ to finish the proof.
