Abstraction from Gauss-Jordan Elimination Method

The method we applied in Section 1.2 to solve the linear system given in Section 1.1.2 is called the Gauss-Jordan Elimination method. The operations we have applied so far are:

1. Multiplying a row by a constant and adding it to another row.
2. Multiplying a row by a non-zero constant.

Before we go any further, let us introduce a new terminology:

Definition 2   (Leading Coefficient)

For a coefficient matrix, each non-zero row's first non-zero entry is called its leading coefficient.

It is important to note that we are not blindly applying arbitrary row operations on a system. Given a linear system, we extract its augmented matrix and simplify this matrix by

(a)

Eliminating as many entries of the coefficient matrix as possible.

(b)

For a non-zero row, making its leading coefficient to be 1.

The objective matrix we are aiming at is the row echelon matrix of the following format:

Definition 3   (Row Echelon Matrix)

A row echelon matrix is a matrix satisfying the following:

1. The leading coefficient of a non-zero row is made to be 1.
2. For a column containing a leading coefficient in it, the rest entries except the leading coefficient one are zeros.

For instance, given the following matrix, we may (1) multiply the $i$-th row by $-a_{12}$ and add it to the first row and (2) multiply the second row by $\frac{1}{a_{2j}}$. The result matrix is then a little closer to its row echelon form.

Given an arbitrary augmented matrix, the reason we try to apply the above two row operations to simplify it to a row echelon matrix is: ``For a row echelon matrix, we may simply read out the solution set for the linear system it represents."