HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

Discussed  short cut for inverse of matrix and solving linear equations of three variables .

Consider three linear equations of three variables .

2x + 3y - 4z = 10

3x - 2y + 4z = 12

  x  -  y  +  z  =  5
Changing these equations into Matrix Form like this

AX = B
X =  A^-1 B - - - - -(1)

where A is Matrix of 3×3 order , which consist of  Coefficients of  x ,y  and z respectively. X is the matrix of order 3×1 and whose elements are  variables  in given linear equations . B is the matrix of 3×1 and consist of  the constant terms on the right hand sides of all the equations such that

How to Find A^-1 

To find A^-1 ,we have to check the determinant value of Matrix A, if the Determinant value of Matrix A is non Zero then  A^-1 Exists, otherwise not.

How to find Determinant  Det |A|

|A| = 2(2) + 3(1) - 4(-1)

|A| = 2(2) + 3(1) - 4(-1)

|A| = 4 + 3 +4

|A| =  7
Since |A| is non Zero therefore A^-1 Exists

To find Co factors of elements  and Ad joint Matrix of A

1st of  all  put all the elements of matrix A in 3 rows and 3 columns as written in matrix A then copy the 1st and 2nd columns as  4th and 5th columns , After this we have 3×5 arrangement as shown is 1st  figure given below, Now  complete the arrangement as 5×5 by copying 1st row and 2nd row as 4th and 5th row respectively. It can be seen in 2nd figure given below.

Find Co factors of A₁₁  element and write it in C₁₁   position

The element whose co factor is to be find out , is marked in red. and the co factor will be calculated by eliminated that row and column in which red coloured element is lying, Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of  A₁₂  element and write it in C₁₂   position

Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of  A₁₃  element and write it in C₁₃   position

Here co factor will be calculated by cross multiplication of  purple coloured four elements

Find Co factors of A₂₁  element and write it in C₂₁   position

Find Co factors of A₂₂  element and write it in C₂₂   position

Find Co factors of  A₂₃  element and write it in C₂₃   position

Find Co factors of A₃₁  element and write it in C₃₁   position

Find Co factors of A₃₁  element and write it in C₃₁   position

Find Co factors of A₃₁  element and write it in C₃₁   position

How to find values of  x , y and z

Therefore ad joint of Matrix A can be written as below

Now using the property of equality of two matrices ,
x   =   52/22   
y   =   -18/11 and 
z    =  -15/11

Verification 

we can check whether the values of x , y and z so calculated satisfies our system of linear equation by putting their values in one or all the given equations.

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Conclusion

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