Echelon Forms

By using pivot positions and basic row operations, the following procedure produces a reduced echelon form (REF).

The size $ k$ of pivot positions used in the above procedure is called the rank of $ A$, denoted by  rank$ A$.

EXAMPLE 1. Apply row operations to reduce the following matrix to an echelon form.

$\displaystyle \left[\begin{array}{rrrrr}
0 & -3 & -6 & 4 & 9 \\
-1 & -2 & -1 & 3 & 1 \\
-2 & -3 & 0 & 3 & -1 \\
1 & 4 & 5 & -9 & -7
\end{array}\right]
$

EXAMPLE 2. Apply row operations to transform the following matrix first into an echelon form, and then reduce it into the reduced echelon form.

$\displaystyle \left[\begin{array}{rrrrrr}
0 & 3 & -6 & 6 & 4 & -5 \\
3 & -7 & 8 & -5 & 8 & 9 \\
3 & -9 & 12 & -9 & 6 & 15
\end{array}\right]
$

Matlab comes with a rref function for row reducing matrices. For Octave you can download rref.m from our web site, and place it into your working directory. Provided an augmented matrix A, type

> rref(A)
which produces a REF.

In Matlab/Octave the function rank(A) can be called to compute the rank of matrix A.



Subsections


Department of Mathematics
Last modified: 2005-10-21