Gaussian-Jordan Elimination
Definition1
An augmented matrix is said to be in echelon form if it satisfies the following conditions:
In rows that have a non-zero element, the first non-zero number is a 1, referred to as the leading 1.
Rows where all elements are zero are placed at the bottom.
For consecutive rows that contain non-zero elements, the leading 1 in the upper row must be to the left of the leading 1 in the row below.
If an echelon form matrix further satisfies the condition below, it is said to be in reduced echelon form:
- In columns that contain a leading 1, all other elements must be zeros.
The following matrices are in reduced echelon form:
$$ \begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & -1 \end{bmatrix},\quad \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\quad \begin{bmatrix} 0 & 1 & -2 & 0 & 1 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix},\quad \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$
The following matrices are in echelon form but not in reduced echelon form:
$$ \begin{bmatrix} 1 & 4 & -3 & 7 \\ 0 & 1 & 6 & 2 \\ 0 & 0 & 1 & 5 \end{bmatrix},\quad \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix},\quad \begin{bmatrix} 0 & 1 & 2 & 6 & 0 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$
The process of using elementary row operations on the augmented matrix of a given linear system to produce a reduced echelon form is referred to as the Gauss-Jordan elimination. The process of making all elements below a leading 1 into zeros is called forward elimination, and making all elements above a leading 1 into zeros is called backward elimination.
Properties
Every matrix has a unique reduced echelon form. In other words, regardless of the sequence of elementary row operations performed, the same reduced echelon form matrix is obtained.
Echelon forms are not unique. That is, different echelon forms can be obtained depending on the sequence of elementary row operations.
The number of rows in an echelon form where all elements are zeros is the same, and the position of the leading 1s is also the same. These positions are referred to as the pivot positions.
General Solution2
When a linear system has infinitely many solutions, a set of parametric equations that generates solutions by substituting parameters is called the general solution of the linear system.
Consider a linear system whose augmented matrix is transformed into the following reduced echelon form by elementary row operations:
$$ \begin{bmatrix} 1 & 0 & 3 & -1 \\ 0 & 1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$
Then, the parametric equations are as follows:
$$ x = -1 -3t,\quad y = 2 + 4t, \quad z = t $$
In this case, variables corresponding to the leading 1 are called leading variables, and the other variables are called free variables.