55. Linear System Special Cases

When you reduce an augmented matrix to REF, rows of all zeros (or zero coefficients with non-zero RHS) signal one of three structural situations.

55.1. 1. Dependent equations

One equation is a linear combination of others — they carry the same information.

The second equation is half the first — same line. After row reduction:

The zero row means one equation was redundant. The system has infinitely many solutions (a whole line in ).

55.2. 2. Underdetermined systems

More variables than independent equations — fewer constraints than unknowns.

Two equations in three unknowns. After row reduction:

Both equations are independent, but is a free variable. Infinitely many solutions (a line in ).

55.3. 3. Inconsistent systems

A zero coefficient row with a non-zero RHS — translates to for , which is impossible.

(Same left-hand side scaled by , but right-hand side doesn’t scale to match.)

The second row reads . No solution.

55.4. See also