AcademyLinear Systems
Academy
Reduced Row Echelon Form
Level 1 - Linear Algebra topic page in Linear Systems.
Principle
Reduced row echelon form, abbreviated RREF, is a stricter version of echelon form. It makes the solution structure of a linear system as explicit as possible.
In RREF, each pivot is a leading \(1\), and each pivot column has zeros everywhere except at that pivot.
Notation
\(\operatorname{rref}(A)\)
the reduced row echelon form of A
\(\text{pivot}\)
a leading non-zero entry in a non-zero row
\(\text{leading 1}\)
a pivot scaled to equal 1
\(\text{free variable}\)
a variable whose column has no pivot
\(\sim\)
row-equivalent
The Core Test
A matrix is in reduced row echelon form when it satisfies echelon form and also:
- Every pivot equals \(1\).
- Each pivot is the only non-zero entry in its column.
Typical RREF shape:
RREF pattern
\[\begin{pmatrix}1&*&0&*\\0&0&1&*\\0&0&0&0\end{pmatrix}\]
Every matrix has exactly one RREF. Different row-reduction paths lead to the same final RREF.
Worked Cases
Question
Is \(\begin{pmatrix}1&0&2&|&3\\0&1&-1&|&4\end{pmatrix}\) in RREF?
Answer
The pivots are the \(1\) in column \(1\) and the \(1\) in column \(2\). They move to the right as rows go down. Each pivot column has zeros everywhere except at the pivot. Therefore the matrix is in RREF. The third variable is free because column \(3\) has no pivot.
Examples
Question
Read the solution from \(\begin{pmatrix}1&0&0&|&2\\0&1&0&|&-1\\0&0&1&|&5\end{pmatrix}\).
Answer
Each variable column is a pivot column. The rows give \(x=2\), \(y=-1\), and \(z=5\). Therefore the system has the unique solution \((2,-1,5)\).