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  
$\displaystyle \vdots$                          $\displaystyle \vdots$   $\displaystyle \vdots$  
am1x1 + am2x2 + … + amnxn = bm  

can be applied equally well to the matrix of coefficients and scalars

$\displaystyle \left[\vphantom{
\begin{array}{ccccc}
a_{11} & a_{12} & \ldots...
...& \vdots \\
a_{m1} & a_{m2} & \ldots & a_{mn} & b_{m}
\end{array}
}\right.$$\displaystyle \begin{array}{ccccc}
a_{11} & a_{12} & \ldots & a_{1n} & b_{1} \...
... & \vdots & \vdots \\
a_{m1} & a_{m2} & \ldots & a_{mn} & b_{m}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{ccccc}
a_{11} & a_{12} & \ldots...
...& \vdots \\
a_{m1} & a_{m2} & \ldots & a_{mn} & b_{m}
\end{array}
}\right]$

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.



Subsections