The question which remains to be solved is: `` are there other solutions?''. First we will see examples.

The restricted matrix of this homogeneous system is
.
We row reduce it:

This last matrix is the restricted matrix of an homogeneous system which has only the trivial solution.

The restricted matrix of this homogeneous system is .

Let us row-reduce the matrix *A*:

The colums corresponding to the unknowns *x*, *y* and *z* contains pivots, thus these unknowns are principal. The last unknown (i.e. *t*) is free; we obtain the principal unknowns as functions of *t*:

The set of solutions of the given system is thus:

We can make a remark, which will be useful for the next chapter: define *t*=37*u*, for
.
Then we have: