RREF

Principle

Reduced row echelon form, or RREF, is a standard final form for a matrix after row reduction. It makes the equations easy to read because pivot columns are cleaned both above and below each pivot.

RREF is unique for a matrix, even though different valid row-operation paths may be used to reach it.

Notation

• A leading entry is the first nonzero entry in a nonzero row.
• In RREF, every leading entry is \(1\).
• A pivot column contains a leading \(1\).
• A pivot variable corresponds to a pivot column.
• A free variable corresponds to a non-pivot variable column.

Method

To read a solution set from RREF:

1. Identify the pivot columns.
2. Treat variables in pivot columns as pivot variables.
3. Treat variables in non-pivot columns as free variables.
4. Read each nonzero row as an equation.
5. Assign parameters to free variables.
6. Write the pivot variables in terms of those parameters.

Rules

• Every nonzero row has leading entry \(1\).
• Each pivot column has zeros above and below its leading \(1\).
• Pivots move to the right as rows go down.
• Zero rows, if any, are at the bottom.
• Free variables may take any real values unless another condition restricts them.

Examples

Suppose the RREF of an augmented matrix for variables \(x\), \(y\), and \(z\) is

\[ \left(\begin{array}{ccc|c} 1&0&2&5\\ 0&1&-1&1\\ 0&0&0&0 \end{array}\right). \]

The pivot columns are the \(x\)-column and the \(y\)-column, so \(x\) and \(y\) are pivot variables. The \(z\)-column is not a pivot column, so \(z\) is free.

Reading the rows gives

\[ \begin{aligned} x+2z&=5,\\ y-z&=1. \end{aligned} \]

Let \(z=t\), where \(t\in\mathbb R\). Then

\[ x=5-2t, \qquad y=1+t, \qquad z=t. \]

The solution set is

\[ (x,y,z)=(5-2t,1+t,t),\qquad t\in\mathbb R. \]

Checks

• Check that each nonzero row has a leading \(1\).
• Check that pivot columns have zeros both above and below each pivot.
• Check that pivots move right as rows go down and that zero rows are at the bottom.
• Check zero rows before deciding which variables are free.
• Check the solution by reading equations from each row, not from columns alone.