Degenerate cases of Gauss

Not all systems have a unique solution. The triangular form obtained with Gauss lets us recognise at a glance the cases in which the solutions are zero or infinite.

Warning — Degenerate cases of Gauss

During elimination there may appear:

• a row $(0,0,\ldots,0\mid b)$ with $b\ne 0$ $\Rightarrow$ inconsistent system;
• a row $(0,0,\ldots,0\mid 0)$ $\Rightarrow$ underdetermined system (infinitely many solutions depending on a parameter).

The triangular form lets us read off the number of solutions at sight: $n$ non-zero rows = unique solution; fewer than $n$ = infinitely many.

Connections

Topics: Linear systems
Concepts: Gauss’s method · Inconsistent system · Underdetermined system
Skills: Reasoning by cases · Solving systems