The matrix $A$ is in the span of $S$, since
\begin{align*}
\begin{bmatrix} -13 & 24 & 2 \\ -8 & -2 & -20 \end{bmatrix}
&=
2\begin{bmatrix} -2 & 3 & 4 \\ -1 & 3 & -2 \end{bmatrix}
-2\begin{bmatrix} 4 & -2 & 2 \\ 0 & -1 & 1 \end{bmatrix}
-3\begin{bmatrix} -1 & -2 & -2 \\ 2 & 2 & 2 \end{bmatrix}
+ 4\begin{bmatrix} -1 & 2 & -2 \\0 & -1 & -2 \end{bmatrix}
\end{align*}
Note that if we were to write a complete linear combination of all of the matrices in $S$, then the fourth matrix would have a zero coefficient.