SOLVING LINEAR SYSTEMS USING MATRICES

The following steps will be useful to solve a system of linear equation using matrices. 

Step 1 : 

Write the given system of linear equations as matrix. 

Example : 

Let us consider the following system of linear equations. 

2x + 3y  =  8

7x - 2y  =  3

The above system can be written as a matrix as shown below. 

Step 2 : 

To solve the given system of linear equations, we have to use row operations and write the above matrix in reduced echelon form. 

Reduced row echelon form for the system of two linear equations :  

From the above matrix, the solution to the system of two linear equations is (a, b).

Reduced row echelon form for the system of three linear equations :  

From the above matrix, the solution to the system of three linear equations is (a, b, c).

Row Operations

Row operations can be applied to create an equivalent matrix. 

There are three basic row operations. 

1.Switching Rows :

Switching rows allows you to switch a row with another row. 

Example :

Switching the first row with the second row. 

2.Multiplication and Division :

Multiply or divide elements of a row by a nonzero constant.  

Example :

* Multiply the first row by 3. 

* Multiply the second row by 1/2, which is equivalent to dividing by 2.   

3.Addition and Subtraction :

Add elements of a row to (or subtract them from) elements of another row. This produces a new row, which you can use to replace one of the two rows you just combined.    

Example :

Subtract the second row from the first row, replacing the first row with the difference.  

Solved Examples

Example 1 : 

Solve the system of linear equations given below using matrices. 

x - 2y  =  25

2x + 5y  =  4

Solution :

Write a matrix representation of the system of equations.

To solve a linear system of equations using a matrix, analyze and apply the appropriate row operations to transform the matrix into its reduced row echelon form.  

The final matrix is in reduced row echelon form and it allows us to find the values of x and y. 

Hence, the solution of the system of linear equations is 

(7, -2)

That is,

x  =  7  and  y  =  - 2

Justification : 

Check that the values x = 7 and y = -2 satisfy both the equations. 

Example 2 : 

Solve the system of linear equations given below using matrices. 

x + y + 2z  =  13

2y + z  =  5

2x - y  =  6

Solution :

Write a matrix representation of the system of equations.

As we have done in example 1, apply the appropriate row operations to transform the matrix into its reduced row echelon form. 

The final matrix is in reduced row echelon form and it allows us to find the values of x, y and z. 

Hence, the solution of the system of linear equations is 

(3, 0, 5)

That is,

x  =  3, y  =  0  and  z  =  5

Justification : 

Check that the values x = 3 and y = 0 and z = 5 satisfy the given system of linear equations. 

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