Assume the following equation holds when $x, a, b,c,d$(variables) taking any values: $$ \begin{bmatrix} r_0&r_1&r_2 \\ s_0&s_1&s_2\\t_0&t_1&t_2 \end{bmatrix} \begin{bmatrix} 1&1 \\ x&2x\\x^2&-x^2 \end{bmatrix} = \begin{bmatrix} a+b&c+d \\ a-b&c-2d\\a+2b&2c+d \end{bmatrix} $$ one can get following constraints on $r_i, s_i, r_i$ by substitutions: $$ \begin{cases} 2t_0-3r_0+s_0=0\\ 2t_1-3r_1+s_1=0\\ 2t_2-3r_2+s_2=0\\ \end{cases} \begin{cases} 3t_0-5r_0-s_0=0\\ 3t_1-5r_1-s_1=0\\ 3t_2-5r_2-s_2=0\\ \end{cases} \\ $$ Thus, the values of $r_i, s_i, r_i$ can be derived by performing pseudoinverse on the matrix equations: $$ \begin{bmatrix} 2 & -3 &1 \\ 3 & -5 &-1 \end{bmatrix} \begin{bmatrix} t_0 \\ r_0 \\ s_0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\\ $$ $$ \begin{bmatrix} 2 & -3 &1 \\ 3 & -5 &-1 \end{bmatrix} \begin{bmatrix} t_1 \\ r_1 \\ s_1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ $$ $$ \begin{bmatrix} 2 & -3 &1 \\ 3 & -5 &-1 \end{bmatrix} \begin{bmatrix} t_2 \\ r_2 \\ s_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ I feel it's an inverse problem of interpolation with Vandermonde matrix.
Is there a well-defined technique to do the above calculations? The matrix equation involved in my work is actually much bigger, thus I am looking for a systematic way to derive $r_i, r_i, t_i $ without given any particular values of $x, a, b,c,d$.