Echelon Form

Definition

A matrix is in echelon form if it is in row echelon form and has all leading entries in the first column, second column, and so on.

The first non-zero element in each row, called the leading entry, is \(1\).
Each leading entry is in a column to the right of the leading entry in the previous row.
Rows with zero elements, if any, are below rows having a non-zero element.
For a non-zero row, the leading entry in the row is the only non-zero entry in the column.

Note

Suppose for some \(i\), \(i^{th}\) row of \(A\) is a zero row but \(b_i \neq 0\). Then the system of equations \(Ax=b\) has no solution.

Reason being that if we write the corresponding system of linear equations, the \(i^{th}\) equation is

\(0x_1 + 0x_2 + \cdots + 0x_n = b_i\). This solution is not possible.

Example

\(A_{ref} = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\)
\(A\) is in row echelon form.
Converting \(A\) to reduced row echelon form:
\(A_{rref} = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\)
\(A\) is in reduced row echelon form.

Let \(Ax=b\) where \(A\) is in the form of \(A_{rref}\). Assume that for every zero row of \(A\), \(b_i = 0\).

If the \(i^{th}\) column has the leading entry of some row, we call \(x_i\) a dependent variable.
If the \(i^{th}\) column has no leading entry, we call \(x_i\) an independent variable.

Example

\(A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\)
\(A\) is in reduced row echelon form.
In this case, \(\mathbf{x_1}\), \(\mathbf{x_3}\) and \(\mathbf{x_4}\) are dependent variables and \(\mathbf{x_2}\) is an independent variable.