determine all possibilities for the solution set infinitely

determine all possibilities for the solution set (infinitely many soolutions; as a unique solution; or no solution). A homogeneous system of 3 equations with 3 unknowns that has solution
x1=1, x2=3, x3=-1

Solution

Since the existence of multiple solutions (provided that there is any solution at all) depends only on the coefficient matrix and since a homogeneous system always has at least one solution (namely the trivial one), multiple solutions for a linear system are possible only if the corresponding homogeneous system has multiple solutions. But the homogeneous system has multiple solutions if and only if it has a non-trivial solution.

Infinitely many solutions. To see why note system is consistent since trivial solution exists. Then at most 3 nonzero rows in (R)EF, as only 3 equations, i.e. r 3.

Thus infinitely many solutions.