Gaussian Elimination and Rank

Skeleton of a System

Gaussian elimination transforms a matrix into row echelon form: a staircase pattern where each row's leading nonzero entry (pivot) sits to the right of the one above it.

Applying row operations (swapping, scaling, adding multiples) preserves solutions while clearing entries below pivots. Reduced row echelon form makes every pivot 1 and the sole nonzero entry in its column.

Pitfall: A zero row ($0=0$) means redundancy and infinitely many solutions. An inconsistent row ($0=1$) means contradiction and no solution.

Rank and Solution Count

The number of pivots is the rank: the count of independent rows. Rank measures genuine information:

• Full rank ($\text{rank} = n$ for $n \times n$): Invertible, unique solution to $A\mathbf{x} = \mathbf{b}$.
• Rank deficiency ($\text{rank} < n$): Dependent rows, free variables, no unique solution.

The number of free variables equals (unknowns $-$ rank). Zero free variables yield a single solution; more yield infinite.

$$\text{Free variables} = n - \text{rank}$$