We start with the linear set of equations $$ \mathbf{A}\mathbf{x} = \mathbf{w}. $$ We assume also that the matrix \( \mathbf{A} \) is non-singular and that the matrix elements along the diagonal satisfy \( a_{ii} \ne 0 \). Simple \( 4\times 4 \) example $$ \begin{bmatrix} a_{11}& a_{12} &a_{13}& a_{14}\\ a_{21}& a_{22} &a_{23}& a_{24}\\ a_{31}& a_{32} &a_{33}& a_{34}\\ a_{41}& a_{42} &a_{43}& a_{44}\\ \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \\ x_4 \\ \end{bmatrix} =\begin{bmatrix} w_1\\ w_2\\ w_3 \\ w_4\\ \end{bmatrix}. $$