Identifying Free Variables in a Homogeneous Matrix

[lom]

$$\begin{pmatrix} 3 &0 &0 &-1 &0 &0 \\ 2& 0& 0 & 0 & 0 &-4 \\ 8&2 &0 &-3 &-1 &0 \\ 0& 1& 0& -1& 0& 0\\ 0& 0 &1 &0 &-1 &0 \end{pmatrix}$$
i got here 5 equations
and 6 variables

how to know who is the free variable
without making a triangle of zeros
on the bottom
?

 

[LCKurtz]

You can't. And there may be more than one free variable. Which variables are free may depend on how your do your row reduction. And the completed row reduction might not come out "triangular".