Say I have $4$ simultaneous equations
\begin{cases} 4.3S_1 - P = T \\ 8S_2 - P = T \\ 5.5S_3 - P = T \\ S_1 + S_2 + S_3 = T. \end{cases}
I'm trying to solve these in Excel using MINVERSE and MMULT it works fine if I build a matrix for just $4$ terms where $T = 10$
$$\begin{bmatrix} 4.3&0 & 0 & -1 & 10 \\ 0 & 8 & 0 & -1 & 10 \\ 0 & 0 & 5.5 & -1 & 10 \\ 1 & 1 & 1 & 0 & 10 \end{bmatrix} $$
I now want to extend this and use any number of $S_1, \ldots, S_x$ terms where some can be zero. If Sx is zero then Excel is forcing me to use a $-1$ in the matrix instead of $0$. Please can someone help me with the maths to verify this is correct (as simply as possible). Why is it $-1$ and not $0$?
E.g., assuming terms $S_1$ to $S_6$ and $S_4 = 0, S_5 = 0, S_6 = 0.$ I would have expected to use the following as an input matrix
$$\begin{bmatrix} 4.3 & 0 & 0 & 0 & 0 & 0 & -1 & 10 \\ 0 & 8 & 0 & 0 & 0 & 0 & -1 & 10 \\ 0 & 0 & 5.5 & 0 & 0 & 0 & -1 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 10 \end{bmatrix} $$
but Excel only works using the following where $S_4 = -1, S_5 = -1, S_6 = -1.$
$$\begin{bmatrix} 4.3 & 0 & 0 & 0 & 0 & 0 & -1 & 10 \\ 0 & 8 & 0 & 0 & 0 & 0 & -1 & 10 \\ 0 & 0 & 5.5 & 0 & 0 & 0 & -1 & 10 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 10 \end{bmatrix} $$
this may help set things up in Excel
Notice that if you enter $0$, you have the equations: $0\cdot S_4 = 0, 0 \cdot S_5 = 0$ and $0\cdot S_6 = 0$. I think Excel does not work with zero maybe to avoid a zero division $(S_4 = \frac{0}{0}?)$. Actually, any non-zero number works, because $(S_4 = S_5= S_6 = \frac{0}{n} =0)$.
Remark: The elements $-1$ on the diagonal does not mean that $S4 =S5=S6=-1$, but that $(-1)S_4 = 0, (-1)S_5 = 0$, and $(-1)S_6 = 0.$