Question

How Do you find free variables in a matrix?

Answer 1

Definition of free and basic Variables:

A variable is a basic variable if it correlated to a pivot column. Else, the variable is known as a free variable. To determine difference between basic and free variable, it is necessary to row reduce the augmented matrix to echelon form.

Steps to Transform a Matrix Into Its Echelon Forms :

1. Identify the last row having a pivot equal to $1$, the pivot row.
2. Add multiples of the pivot row to each of the upper rows, until every element above the pivot equals $0.$
3. Move up the matrix, repeat this process for each row.

Example to identify the free and basic variable :

Let the system of linear equations

$\begin{array}{rcl}{x}_1+2x_2-x_3& =& 4\\ 2x_1-4x_2& =& 5\end{array}$

This system has an augmented matrix. let it be $A$ which is given as

$A=\left[\begin{array}{cccc}1& 2& -1& 4\\ 2& -4& 0& 5\end{array}\right]$

Now, $R_2\to R_2-2R_1$ which row $R_2$ reduces to

$A=\left[\begin{array}{cccc}1& 2& -1& 4\\ 0& -8& 2& -3\end{array}\right]$

matrix $A$ is in the echelon form, so we can identify the pivot positions.

The first and second columns are pivot columns, so variables $x_1$ and $x_2$are basic variables.

the $3rd$ column is not a pivot column, So $x_3$ is a free variable.

Hence, in this example variables $x_1$ and $x_2$are basic variables and $x_3$ is a free variable.