AcademyLinear Systems
Academy
Echelon Form
Level 1 - Linear Algebra topic page in Linear Systems.
Principle
Echelon form is a stepped form of a matrix that makes a linear system easier to solve. The leading non-zero entries move strictly to the right as you go down the rows, and any zero rows sit at the bottom.
Echelon form is the target of forward elimination.
Notation
\(R_i\)
row i of a matrix
\(\text{pivot}\)
the first non-zero entry in a non-zero row
\(\text{pivot column}\)
a column containing a pivot
\(\text{zero row}\)
a row whose entries are all zero
\(\sim\)
row-equivalent
The Core Test
A matrix is in echelon form when all three conditions hold:
- Every zero row is below every non-zero row.
- Each pivot is to the right of the pivot in the row above it.
- Every entry below a pivot is zero.
The pivots create a staircase pattern. For example,
Echelon pattern
\[\begin{pmatrix}*&*&*&*\\0&*&*&*\\0&0&0&*\\0&0&0&0\end{pmatrix}\]
where each displayed \(*\) pivot position is non-zero.
Worked Cases
Question
Is \(\begin{pmatrix}1&2&0&|&3\\0&0&5&|&1\\0&0&0&|&0\end{pmatrix}\) in echelon form?
Answer
The first pivot is in column \(1\). The second pivot is in column \(3\), which is to the right of column \(1\). The zero row is at the bottom. Entries below the pivots are zero. Therefore the matrix is in echelon form.
Examples
Question
Solve the echelon system represented by \(\begin{pmatrix}1&2&-1&|&4\\0&1&3&|&5\\0&0&2&|&6\end{pmatrix}\).
Answer
Translate rows to equations: \(x+2y-z=4\), \(y+3z=5\), and \(2z=6\). From the last row, \(z=3\). Substitute into the second row: \(y+3(3)=5\), so \(y+9=5\), giving \(y=-4\). Substitute into the first row: \(x+2(-4)-3=4\), so \(x-11=4\), giving \(x=15\).