Free And Basic Variables Of A Matrix - Examples With Solutions

Free and Basic Variables of a Matrix - Examples with Solutions

Free and Basic Variables of a Matrix Representing a System of Equations

Definition For a system of equations in row echelon form , which of course may be represented by the augmented matrix, a variable whose coefficient is a leading 1 ( pivot ) is called a basic variable and a varibale without pivot is called a free variable. Let us consider the following system of equations in row echelon form The augmented matrix in row echelon of the above system is as follows According fo the above definition x1, x3 and x4 are the basic variables and x2 and x5 are the free variables. When we solve the above system, we express the basic variables in terms of the free variables The third equation in the system gives x4 = x5 The second equation gives x3 = 2 x5 The first equation gives x 1 = − 2 x 2 + x 3 − x 4 + 3 x 5 Substitute the basic variables on the right x 1 = − 2 x 2 + 2 x 5 − x 5 + 3 x 5 Simplify x 1 = − 2 x 2 + 4 x 5 The basic variables are written in terms of the free variables as x 1 = - 2 x 2 + 4 x 5 x 3 = 2 x 5 x 4 = x 5 where x 2 and x 5 can be any real numbers hence their names as "free variables". Definition The use of free variables helps us to write an explicit formula for the solutions of our system.

Questions with Solution

For each of the following augmented matrices in row echelon form, which are basic variables and which are free variables?

1. [ 1 4 3 0 0 0 0 1 -3 1 0 0 0 0 0 ]
2. [ 1 0 0 0 1 0 ]
3. [ 1 2 0 -1 -2 0 0 1 1 0 3 0 0 0 0 1 1 0 0 0 0 0 0 0 ]
4. [ 1 2 0 0 0 0 1 0 0 0 0 0 ]

Solutions to the Above Questions

Being augmented matrices, the number of variables is equal to the number of columns of the given matrix -1. For examples, for a matrix of 5 columns, the number of variables is 5 - 1 = 4, named as x1, x2, x3 and x4.

1. Matrix 1 is has two pivots and 4 variables. The first pivot at row 1 column 1; hence x1 is a basic variable. The second pivot is at row 2 column 3; hence x3 is also a basic variable. The remaining variables: x2 and x4 are free variables.
2. Matrix 2 is has two Pivots and 2 variables. The first pivot is at row 1 column 1; hence hence x1 is a basic variable. The second pivot is at row 2 column 2; hence x2 is a basic variable. There are no free variables.
3. Matrix 3 has 3 pivots and 5 variables. The first pivot at row 1 column 1; hence x1 is a basic variable. The second pivot is at row 2 column 2; hence x2 is a basic variable. The third pivot is at row 3 column 4; hence x4 is a basic variable. The remaining variables: x3 and x5 are free variables.
4. Matrix 4 has two Pivots and 3 variables. The first pivot is at row 1 column 1; hence hence x1 is a basic variable. The second pivot is at row 2 column 3; hence x3 is a basic variable. The remaining variables: x2 is a free variables.

More References and links

1. Pivots of a Matrix in Row Echelon Form
2. linear algebra
3. Solve a system of linear equations by elimination
4. elementary matrices