Answer
$C$ is not in the span of $A$ and $B$.
Work Step by Step
For $C$ to be in the span of $A,B$, there must exist $a,b \in \mathbb{R}$ such that $aA+bB = C$. So, find a solution to
$$\begin{bmatrix}
a & a \\
-a & a
\end{bmatrix} + \begin{bmatrix}
b & -b \\
b & 0
\end{bmatrix} = \begin{bmatrix}
a+b & a-b \\
b-a & a
\end{bmatrix} = \begin{bmatrix}
1 & 2 \\
3 & 4\\
\end{bmatrix}$$
However, this would mean that $a-b = 2$ and $b-a = 3$. This is impossible. So, the matrix $C$ is not in the span of $A$ and $B$.
