Row Operations and Echelon Forms
One of the elementary
applications of matrix arrays is
storing and
manipulating coefficients of systems of linear equations. The various steps
that you carry out in applying the technique of elimination to a system of
linear equations
a11x1 + a12x2 + … + a1nxn |
= |
b1 |
|
a21x1 + a22x2 + … + a2nxn |
= |
b2 |
|
|
 |
 |
|
am1x1 + am2x2 + … + amnxn |
= |
bm |
|
can be applied equally well to the matrix of coefficients and scalars
For this and numerous other reasons, you do elementary row operations on matrices. The goal of elementary row operations is to put the matrix in
a special form, such as a row echelon form, where the number of
leading zeroes increases as the row number increases. The system provides
several choices for obtaining a row echelon form
, one of
which gives the reduced row echelon form satisfying the following
conditions.
- The number of leading zeroes increases as the row number increases.
- The first nonzero entry in each nonzero row is equal to 1.
- Each column that contains the leading nonzero entry for any row
contains only zeroes above and below that entry.
Subsections