# Using ZeroTest method with RowReduce

My goal is to solve this system:

\begin{align*} x_1+2x_2+2x_3+2x_4&=a\\ 2x_1+4x_2+6x_3+8x_4&=b\\ 3x_1+6x_2+8x_3+10x_4&=c \end{align*}

Setting up the augmented matrix and using RowReduce,

mat = {{1, 2, 2, 2, a}, {2, 4, 6, 8, b}, {3, 6, 8, 10, c}};
MatrixForm[RowReduce[mat]]


produces:

\begin{bmatrix} 1 & 2 & 0 & -2 & 0\\ 0 & 0 & 1 & 2 & 0\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}

A first glance at the last row indicates there are no solutions, which is not true. There are either no solutions or an infinite number of solutions depending on the values of a, b, and c.

To let my colleagues know, I can solve this using Mathematica. I use elementary row operations to put the matrix in row echelon form, then I can see that there will be solutions if $a+b-c=0$. Then I can pick a particular solution, then find the null space of the coefficient matrix to determine all possible solutions if $a+b-c=0$.

Now, here comes the real question that I need help with. There is a very interesting answer by @ssch here. It uses RowReduce and the option ZeroTest. So I went to the documentation on RowReduce, read that RowReduce[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero. However, there is not even a single example of its use in the RowReduce documentation page.

Now, here is @ssch possible use for this question:

MatrixForm[RowReduce[mat, ZeroTest -> (! FreeQ[#, a] &)]] // Simplify


This remarkably produces the following result:

\begin{bmatrix} 1 & 2 & 0 & -2 & 3a-b\\ 0 & 0 & 1 & 2 & \frac12(-2a+b)\\ 0 & 0 & 0 & 0 & -2(a+b-c) \end{bmatrix}

It doesn't matter if I use FreeQ[#,a], FreeQ[#,b], or FreeQ[#,c]. All three deliver the same result. Now, I did look up FreeQ and I understand its use, but I still don't understand what is going on with the ZeroTest due to the lack of examples in the documentation.

Can some of my colleagues walk me through this with some examples and quality explanation? Thanks.

mat = {{1, 2, 2, 2}, {2, 4, 6, 8}, {3, 6, 8, 10}};