Matrix Multiplication

Firstly, since the product $\begin{pmatrix} 4 \\ 1 \\ 3\end{pmatrix}\times A$ is a $3\times3$ matrix, $A$ must be a $1\times3$ matrix. Thus we have $\begin{pmatrix} 4\\1\\3 \end{pmatrix} \times \begin{pmatrix}a_1 & a_2 & a_3\end{pmatrix} = \begin{pmatrix} 4a_1 & 4a_2 & 4a_3 \\ 1a_1 & 1a_2 & 1a_3 \\ 3a_1 & 3a_2 & 3a_3 \end{pmatrix} = \begin{pmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{pmatrix}$

Since these two matrices are equal, their entries must be equal. Choosing the second row (any row will suffice), $a_1 = -1$ , $a_2 = 2$ and $a_3 = 1$ . Thus $a_1+a_2+a_3 = \boxed{2}$ .