What echelon form reveals

Rank, pivots and the number of solutions

Once the system is in echelon form, its fate can be read from its shape. The key word is rank: the number of pivots.

Counting the pivots

The rank of a system is the number of non-zero rows in its echelon form — in other words the number of pivots. It is the number of genuinely independent equations: those that carry new information.

[ 1   2   1 |  8 ]   <- pivot
[ 0   1   1 |  5 ]   <- pivot            rank = 2
[ 0   0   0 |  0 ]   <- empty row (that equation was redundant)

A row that has become entirely zero means one of the original equations was a combination of the others: it said nothing new.

The decision tree

With n unknowns and rank r, everything comes down to two questions:

                A row "0 = c" with c ≠ 0 ?
                        /              \
                     YES               NO
                      |                  |
                NO solution        compare r and n
              (inconsistent)         /          \
                                  r = n        r < n
                                    |             |
                             ONE solution   INFINITELY MANY
                                            (n - r free parameters)

The number n - r has a name: the degree of freedom of the system, the number of unknowns that can be chosen arbitrarily.

Basic and free unknowns

When r < n, we distinguish:

the basic unknowns: those carrying a pivot;
the free unknowns: the others, taken as parameters.

[ 1   2   0   3 |  5 ]        pivots in columns 1 and 3
[ 0   0   1  -1 |  2 ]
[ 0   0   0   0 |  0 ]

x1, x3 : basic            x2, x4 : free        n - r = 4 - 2 = 2

The basic unknowns are then expressed in terms of the free ones:

x3 = 2 + x4
x1 = 5 - 2x2 - 3x4

The solution set is described with two parameters, say x2 = s and x4 = t.

Too many equations, too few equations

Two useful intuitions, as long as they are not mistaken for absolute rules:

fewer equations than unknowns (m < n): the rank is at most m, so r < n. Such a system, if it has a solution at all, necessarily has infinitely many. It can never have a unique solution.
more equations than unknowns: the system is called overdetermined. It is often inconsistent — the typical case with experimental measurements, which are then handled by least squares rather than by seeking an exact solution.

Careful: these rules are about the rank, not the number of rows written down. Three equations, two of which are identical, have rank only 2.

Summary