Lesson 11: Row Echelon Form, Rank & Solution Classification

1. Row Echelon Form (REF)

A matrix is in row echelon form if:

• All zero rows are at the bottom
• Leading entry (pivot) in each non-zero row is 1 and is to the right of the pivot above
• All entries below each pivot are zero

Example (REF):

\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \end{pmatrix} \]

2. Rank of a Matrix

The rank is the number of non-zero rows in its row echelon form (or number of pivots).

Rank tells us:

• Dimension of column space / row space
• Number of independent equations / variables

Example: rank 2 matrix in 3D → spans a plane (dimension 2).

3. Classifying Solutions Using Rank

For system Ax = b (m equations, n variables):

• Rank(A) = rank([A|b]) → consistent
• Rank(A) = n → unique solution
• Rank(A) < n → infinite solutions (free variables = n - rank)
• Rank(A) < rank([A|b]) → no solution