Solving Systems of Linear Equations: Row Echelon Form and Rank
linear-algebra
Row Echelon Form (REF)
A matrix is in row echelon form when:
- All zero rows are at the bottom
- The leading entry (pivot) of each non-zero row is to the right of the pivot in the row above
- All entries below a pivot are zero
\[\begin{pmatrix} 2 & 1 & -1 & 8 \\ 0 & \frac{1}{2} & \frac{1}{2} & 1 \\ 0 & 0 & -1 & 1 \end{pmatrix}\]
Reduced Row Echelon Form (RREF)
Additionally requires:
- Each pivot is 1
- All entries above a pivot are also zero
\[\begin{pmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 \end{pmatrix}\]
This directly reads off the solution: \(x=2, y=2, z=-1\).
Rank
The rank of a matrix \(A\), denoted \(\text{rank}(A)\), is the number of pivots in its REF — equivalently, the number of linearly independent rows (or columns).
| Condition | Solution |
|---|---|
| \(\text{rank}(A) = \text{rank}([A\|\mathbf{b}]) = n\) | Unique solution |
| \(\text{rank}(A) = \text{rank}([A\|\mathbf{b}]) < n\) | Infinitely many solutions |
| \(\text{rank}(A) < \text{rank}([A\|\mathbf{b}])\) | No solution |
Example
\[A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}\]
Row 2 is \(2 \times\) Row 1, so after elimination it becomes zero. \(\text{rank}(A) = 2\).