Determine whether a particular set of matrices is a subspaces of $M_{2\times2}$

Question

Determine which of the following are subspaces of $M_{2\times2}$. Give reasons.

• (a) $W$, the set of matrices $A$ for which $A$ multiplied by $\begin{bmatrix}1&1\\1&1\end{bmatrix}$ is equal to $\begin{bmatrix}0&0\\0&0\end{bmatrix}$.

Any help with how this could be closed under addition or scalar multiplication would help.

Answer 1

HINT Let's do addition.

Let $A,B \in M_{2 \times 2}$ such that $A E = O$, where $E$ and $O$ are matrices of ones and zeros, respectively.

Then $$ (A+B)E = AE + BE = O + O = O $$

Answer 2

Hint The map $M_{2 \times 2} \to M_{2 \times 2}$ defined by $$A \mapsto \pmatrix{1&1\\1&1} A$$ is linear.

By definition, $W = \ker A$ and hence is a subspace of $M_{2 \times 2}$.

(In the problem statement it's ambiguous whether this map is actually left or right multiplication by the given fixed matrix, but this argument works just as well in both cases.)