I have a system of 6 equations with 6 unknowns that can be written in matrix form below: $$ \left[ \begin{array}{cccccc} a_1 & a_2 & a_3 & 0 & 0 & 0\\ 0 & 0 & 0 & a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 & 0 & 0 & 0\\ 0 & 0 & 0 & b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 & 0 & 0 & 0\\ 0 & 0 & 0 & c_1 & c_2 & c_3\\ \end{array}\right].\left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \end{array}\right]=\left[ \begin{array}{c} r_1 \\ r_2 \\ s_1 \\ s_2 \\ t_1 \\ t_2 \end{array}\right] $$
Where $a,b,c,r,s,t$ are all known.
I can just invert this 6x6 matrix, but that would be a huge pain. Are there any structures/symmetry I can take advantage of here?
I noticed that I can "squash" the matrix like this: $$ \left[ \begin{array}{cccccc} a_1 & a_2 & a_3 & a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 & b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 & c_1 & c_2 & _3\\ \end{array}\right].\left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \end{array}\right]=\left[ \begin{array}{c} r_1 + r_2 \\ s_1 + s_2 \\ t_1 + t_2 \end{array}\right] $$
This reduces the matrix size by a bit, but as far as I know a non square matrix does not have an inverse, so I'm not sure how to proceed from here?