3. Row Reduction and Echelon Forms
2026-01-26
Some Definitions
Nonzero Row or Column
A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry.
Leading Entry
A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).
Echelon Form (阶梯形)
REF and RREF
A rectangular matrix is in row echelon form (REF) if it has the following properties:
- All nonzero rows are above any rows of all zeros.
- 0302−10400 satisfies this property.
- 010024035 does not satisfy this property.
- Each leading entry of a row is in a column to the right of the leading entry of the row above it.
- 100200040356 satisfies this property.
- 040250367 does not satisfy this property.
- All entries in a column below a leading entry are zeros.
- In fact, any matrix that satisfies property 2 automatically satisfies this property.
If a matrix in echelon form satisfies the following additional conditions, then it is in reduced (row) echelon form (RREF):
- The leading entry in each nonzero row is 1.
- Each leading 1 is the only nonzero entry in its column.
- 1000510032000010 does NOT satisfies this property.
Theorem
Each matrix is row equivalent to one and only one reduced echelon matrix (RREF).
Pivot
- A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form A.
- A pivot is a nonzero number in a pivot position.
- A pivot column is a column of A that contains a pivot position.
Solutions of Linear Systems
Basic and Free Variables
Take the following linear system as an example:
100010−510140⎩⎨⎧x1−5x2+x3x30=1=4=0
- The variables x1 and x2 corresponding to the pivot columns in the matrix are called basic variables.
- The other variable, x3, is called a free variable.
It is possible to rewrite the basic variables depending on the free variables:
⎩⎨⎧x1x2x3=1+5x3=4−x3 is free
Consistence
A linear system is consistent if and only if the rightmost column of the augmented matrix is NOT a pivot column.
In other words, a linear system is inconsistent if and only if the augmented matrix has a row of the form [00⋯0b], where b is a nonzero number.
If a linear system is consistent, then it has either:
- a unique solution (if there are no free variables), or
- infinitely many solutions (if there is at least one free variable).