Simplest and shortest Matrix method to solve linear equations of 3 variables

Matrix method to solve linear equations of 3 variables

In this post we are going to understand the concept of solving linear equations of three variables with the help of matrix method .

Matrix method to solve linear equations of three variables with the help of example. 

Set of given equations are 

x - y + z = 2    

2x - y = 0  

 2y - 2 z = 1  

Rearranging these equations in symmetrical form. It means if any one of the variable in any equation is missing then write that missing variable/s with zero coefficient. As we see in this case the coefficient of z in 2nd equation and the coefficient of x in 3rd equation are missing. So we have to write coefficient of x in 2nd equation and coefficient of x in 3rd equation as zero.

x - y + z = 2 ----------------------> (1)

2x - y + 0z = 0 ------------------> (2)

0.x + 2y - 2z = 1 ----------------> (3)

The system of these equations can be transformed into Matrix form .

AX = B , ⇒ X = A^-1B  -------> (*)

Where A is matrix written from the coefficients of x, y and z when these equations are in symmetric form and B is the matrix written from constants from right hand sides in column form and X is matrix of all the variables in column form.

In order to find the solution of set of these equations , first we have to find the inverse of matrix A if it exist then we can find the solution otherwise Matrix method fails to find the solution of the set of linear equations . 

Evaluation of Determinant 

                                                  

|A| = 4 (18 - 4) -0(9 - 3) +6(12 - 18)

       = 4(14) + 0 + 6(-6) 

       = 56 - 36

        = 20

Since the determinant value of this matrix is not equal to zero ,Therefore its inverse can be calculated.

And formula for finding the inverse of matrix A is

Where Adjoint A is the transpose of co factor matrix. And in order to find the co factor matrix of any matrix, we have to find co factors of all the elements present in this matrix. 

How to calculate  co factors of all the elements of the matrix A

Let us calculate these cofactors.  

Now these co factors can be written in matrix form known as co factor Matrix. 

Co factors of 1st row are  (18 - 4) , -(9 - 3), (12 - 18) 

i. e. Co factors of 1st row are 14, -6 , -6

Co factors of 2nd row are -(0 -24), (12 - 18) , -(16 - 0) 

I. e. Co factors of 2nd row are  24 , -6 , -16

Co factors of 3rd row are  (0 - 36), -(4 - 18), (24 - 0) 

i.e. Co factors of 3rd row are  -36 , 14 , 24

Co factor Matrix

Writing co factors of 1st row in 1st row of this matrix , co factors of 2nd row in 2nd row of this matrix . Similarly co factors of 3rd row in 3rd row of this matrix . 

 

Adjoint  Matrix

To find the Ad joint of this matrix we have to take it's transpose, Because transpose of any matrix is called Ad joint of the matrix. So writing all the elements which are in 1st row in 1st column, and  all the elements which are in 2nd row in 2nd column and  all the elements which are in 3rd row in 3rd column. 

Inverse  Matrix

Now we can find inverse of the matrix A by putting the value of inverse of A in equation  (4), Now  putting the values  Matrix B and    A^-1  in (4) After simplification and using the properties of equality of two matrices  ( Two matrices of same order are equal iff their respective elements are equal to each other ) 

Now we shall use the property of equality of two matrices ,which says that if two matrices are equal to each other then their respective elements must be  equal to each other.

⇒ x = 10

    y = 10

    z = 10

So this was the Matrix method of solving linear equations of three variables using inverse of matrix. Your valuables comments will be appreciated for betterment of this blog.