System of Linear Equations in Matrices

In maths, a system of the linear system is a set of two or more linear equation involving the same set of variables. For example : 2x – y = 1, 3x + 2y = 12 . It is a system of two equation in the two variables that is x and y which is called a two linear equation in two unknown x and y and solution to a linear equation is the value to the variables such that all the equations are fulfilled.

In the matrix, every equation in the system becomes a row and each variable in the system becomes a column and the variables are dropped and the coefficients are placed into a matrix.

A system of two linear equations in two unknown x and y are as follows: 

Let  ,  , .

Then system of equation can be written in matrix form as: 

=  i.e. AX = B and  X = .

If the R.H.S., namely B is 0 then the system is homogeneous, otherwise non-homogeneous.

is a homogeneous system of two eqations in two unknowns x and y.

is a non-homogenoeus system of equations.

A system of three linear equations in three unknown x, y, z are as follows:  

.

Let , , . 

Then system of equation can be written in matrix form as: 

=  i.e. AX = B and  X = .

Algorithm to solve the Linear Equation via Matrix

1. Write the given system in the form of matrix equation as AX = B.
2. Find the determinant of the matrix. If determinant |A| = 0, then does not exist so that solution does not exist. Write “System is not consistent”.
3. If the determinant exist then find the inverse of the matrix i.e. .
4. Find where  is the inverse of the matrix.
5. Solve the equation by the matrix method of linear equation with the formula  and find the values of x,y,z.

Example 1: Solve the equation: 4x+7y-9 = 0 , 5x-8y+15 = 0

Solution: Given equation can be written in matrix form as : ,  , 

Given system can be written as : AX = B , where .

Let us find determinant : |A| = 4*(-8) – 5*7 = -32-35 = -67 So, solution exist.

Minor and Cofactor of matrix A are :  = -8  = -8,  = 5 = -5,  = 7 = -7,  = 4 = 4.

Cofactor matrix =   and Adj A = 

  .

=  =  = 

x =  and y = 

Example 2: Solve the equation: 2x+y+3z = 1, x+z = 2, 2x+y+z = 3

Solution: Given equation can be written in matrix form as : , , .

Given system can be written as : AX = B , where .

Let us find determinant : |A| = 2(0-1) – 1(1-2) + 3(1-0) = -2+1+3 = 2. So, solution exist.

Minor and Cofactor of matrix A are :  = -1  = -1,  = -1 = 1, = 1 = 1, = -2 = 2,  = -4 = -4, = 0 = 0 = 1 = -1,  = -1 = -1, = -1 = 1.

and 

  .

=  =  =  = .

x = 3, y = -2, z = -1.

Exercise

Solve the following equations:

1. 2x+3y=9, -x+y=-2.
2. x+3y=-2, 3x+5y=4.
3. x+y=1, 3y+3z=5, 3z+3x=4.
4. x+y+z=1, 2x+y+2z=3, 3x+3y+4z=4.
5. x+y+z=6, 3x-y+3z=10, 5x+5y-4z=3.