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04. Row Reduction and Echelon Forms

Dated: 11-03-2025

Echelon Form of a Matrix

A rectangular matrix1 is in echelon form if it has following properties.

  1. All non zero rows are above any row with all zeros.
  2. As we progress down the rows, each leading entry1 is at the right column.
  3. All entries in a column below a leading entry1 are zero.

Reduced Echelon Form of a Matrix

  • Leading entry1 in each non zero row is \(1\).
  • Each leading \(1\) is the only nonzero entry1 in its column.

Examples of Echelon Form of a Matrix

If \(\cdot\) is the leading entry1 and \(*\) is any number then

\[ \begin{bmatrix} \cdot & * & * & *\\ 0 & \cdot & * & *\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 & 2 & 6 & 0\\ 0 & 0 & 1 & -1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \]

Examples of Reduced Echelon Form of a Matrix

\[ \begin{bmatrix} 1 & 0 & 0 & *\\ 0 & 1 & 0 & *\\ 0 & 0 & 1 & * \end{bmatrix} \]
\[ \begin{bmatrix} 1 & 0 & * & *\\ 0 & 1 & * & *\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{bmatrix} \]

A matrix1 can be row reduced into multiple different echelon forms but only into one unique reduced echelon form.

Theorem 1

Each matrix1 is row equivalent2 to only one and unique reduced echelon matrix.

Pivot Positions

A pivot position in a matrix1 \(A\) is a location in \(A\) that corresponds to a leading entry in an echelon form of \(A\).

When row operations on a matrix1 produce an echelon form, further row operations to obtain the reduced echelon form do not change the positions of the leading entries.

Pivot Column

A pivot column is a column of \(A\) that contains a pivot position.

Example

Reduce \(A\) to echelon form and locate the pivot columns.

\[ A = \begin{bmatrix} 0 & -3 & -6 & 4 & 9 \\ -1 & -2 & -1 & 3 & 1 \\ -2 & -3 & 0 & 3 & -1 \\ 1 & 4 & 5 & -9 & -7 \end{bmatrix} \]

Solution

The pivot position in first row is \(a_{11}\) but it is zero. It has to be a nonzero value so we will replace \(R_1\) with \(R_4\).

\[ A = \begin{bmatrix} 1 & 4 & 5 & -9 & -7 \\ -1 & -2 & -1 & 3 & 1 \\ -2 & -3 & 0 & 3 & -1 \\ 0 & -3 & -6 & 4 & 9 \\ \end{bmatrix} \]

Therefore, \(C_1\) is the pivot column.
All the entries1 below \(a_{11}\) in \(C_1\) should be \(0\). After doing \(R_2 = R_2 + R_1\) and \(R_3 = R_3 + 2 R_1\), we have

\[ A = \begin{bmatrix} 1 & 4 & 5 & -9 & -7 \\ 0 & 2 & 4 & -6 & -6 \\ 0 & 5 & 10 & -15 & -15 \\ 0 & -3 & -6 & 4 & 9 \\ \end{bmatrix} \]
\[\Big \downarrow R_3 = -\frac 5 2 R_2 + R_3\]
\[\Big \downarrow R_4 = \frac 3 2 R_2 + R_4\]
\[ A = \begin{bmatrix} 1 & 4 & 5 & -9 & -7 \\ 0 & 2 & 4 & -6 & -6 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -5 & 0 \\ \end{bmatrix} \]

After interchanging \(R_3\) with \(R_4\), we have

\[ A = \begin{bmatrix} 1 & 4 & 5 & -9 & -7 \\ 0 & 2 & 4 & -6 & -6 \\ 0 & 0 & 0 & -5 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \]

This shows that \(C_1, C_2\) and \(C_3\) are the pivot columns and \(a_{11}, a_{22}\) and \(a_{34}\) are the pivot positions.

Pivot Element

A pivot is a nonzero number in a pivot position that is used as needed to create zeros via row operations

Example

\[ \begin{bmatrix} 1 & 6 & 2 & -5 & -2 & -4\\ 0 & 0 & 2 & -8 & -1 & 3\\ 0 & 0 & 0 & 0 & 1 & 7 \end{bmatrix} \]
\[\Big \downarrow R_1 = R_1 + 2 R_3\]
\[\Big \downarrow R_2 = R_2 + R_3\]
\[ \sim \begin{bmatrix} 1 & 6 & 2 & -5 & 0 & 10\\ 0 & 0 & 2 & -8 & 0 & 10\\ 0 & 0 & 0 & 0 & 1 & 7 \end{bmatrix} \]
\[\Big \downarrow R_2 = \frac 1 2 R_2\]
\[ \sim \begin{bmatrix} 1 & 6 & 2 & -5 & 0 & 10\\ 0 & 0 & 1 & -4 & 0 & 5\\ 0 & 0 & 0 & 0 & 1 & 7 \end{bmatrix} \]
\[\Big \downarrow R_1 = R_1 - 2R_2\]
\[ \sim \begin{bmatrix} 1 & 6 & 0 & 3 & 0 & 0\\ 0 & 0 & 1 & -4 & 0 & 5\\ 0 & 0 & 0 & 0 & 1 & 7 \end{bmatrix} \]

The associated linear system3 is

\[ \begin{aligned} x_1 + 6x_2 + 3x_4 = 0\\ x_3 - 4x_4 = 5\\ x_5 = 7 \end{aligned} \]
\[ \begin{cases} x_1 = -6x_2 - 3x_4\\ x_2 \text{ is free}\\ x_3 = 5 + 4x_4\\ x_4 \text{ is free}\\ x_5 = 7 \end{cases} \]

\(x_2\) and \(x_4\) being free means we can choose any value we want.

References

Read more about notations and symbols.

  1. Read more about matrix. ↩↩↩↩↩↩↩↩↩↩↩

  2. Read more about row equivalence. ↩

  3. Read more about linear system. ↩