This is not related to the issue you have, but you can simplify the notation for these matrices:
$$
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
a_{41} & a_{42} & a_{43} \\
\end{bmatrix}
$$
$$
\begin{pmatrix}
b_1 \\ b_2 \\ b_3 \\ b_4
\end{pmatrix}
$$