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Question

How Do you find free variables in a matrix?


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Solution

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

x1+2x2-x3=42x1-4x2=5

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

A=12-142-405

Now, R2R2-2R1 which row R2 reduces to

A=12-140-82-3

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

The first and second columns are pivot columns, so variables x1 and x2are basic variables.

the 3rd column is not a pivot column, So x3 is a free variable.

Hence, in this example variables x1 and x2are basic variables and x3 is a free variable.


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