HOW TO SOLVE LINEAR EQUATIONS OF THREE VARIABLES BY MATRIX METHOD

How to find solutions of system of linear equations of three variables with  the help of Matrix Method. In this post we shall  solve such types of system of linear equations in which variables lies in the denominators.  Let us solve three linear equations of three variables given below by Matrix Method.

These are not simple linear equations to solve. But these equations can be made simple by following substitution .

After substitution , these equations can be  reduced to following simple linear equations

2x + 3y + 4z = 3

3x - 4y + 5z = 5

  x + 2y - 3z = 6

To solve these equations by Matrix Method , Let us transform this set of linear equations to Matrix form

AX = B, and  X =A^-1 B -----> (4)

where A is matrix comprises the coefficients  x , y and z respectively as shown in the matrix given below , And B is the matrix consist of constant terms on right hand side of these equations and X is the matrix of variables in the linear equations.

1st of all we have to find A^{-1  and then product of the matrices} A^-1 and B  as mentioned in equation (4). But to find inverse of matrix A, its determinant value must be non zero   ( Note it) .

Let us find determinant value of Matrix A as follows :-

|A| = 2[(-4)(-3)–(5)(2)]-3[(3)(-3)-(5)(1)]+4[(3)(2)-(-4)(1)] 

|A| = 2[12-10]-3[-9-5]+4[6+4] 

|A| = 2[2] -3[-14]+4[10] 

|A| = 4 +42+40 

|A| = 86 

Since the value of Determinant of Matrix a is non zero.

  ⇒    A^-1  exists .

To find inverse ,we have to find ad joint of Matrix A . And to find Ad joint of any Matrix , we have to find Co factor Matrix . 

What is Co Factor Matrix

Co factor Matrix can be obtained by the co factors of all the elements of Matrix written in same place  where the element was written originally. 
Formula for Co factor of any element 

        C_ij = (-1)^i+jM_ij 

Co factor of A₁₁ 

Co factor of A₁₁   element can be calculated by eliminating 1st row and 1st column and solving the remaining determinant.

= (-1)¹⁺¹M₁₁ 

= 12 - 10 = 2

Co factor of A₁₂ 

Co factor of A₁₂   element can be calculated   by eliminating 1st row and 2nd column and solving the remaining determinant.

= (-1)¹⁺²M₁₂ 

= -1(-9 - 5 )= 14

Co factor of A₁₃ 

Co factor of A₁₃   element  can be  calculated   by eliminating 1st row and 3rd column and solving the remaining determinant.

= (-1)¹⁺³M₁₃ 

= 6 + 4 = 10

Co factor of A₂₁ 

Co factor of A₂₁   element can be calculated   by eliminating 2nd row and 1st column and solving the remaining determinant.

= (-1)²⁺¹M₁₁ 

= -1(- 9 - 8) = 17

Co factor of A₂₂ 

Co factor of A₂₂   element  can be calculated   by eliminating 2nd row and 2nd column and solving the remaining determinant.
= (-1)²⁺²M₂₂ 

= - 6 - 4 = -10

Co factor of A₂₃ 

Co factor of A₂₃   element can be calculated   by eliminating 2nd row and 3rd column and solving the remaining determinant.

= (-1)²⁺³M₂₃ 

= -1(4 - 3) = -1

Co factor of A₃₁

Co factor of A₃₁   element can be calculated   by eliminating 3rd row and 1st column and solving the remaining determinant.

= (-1)³⁺¹M₃₁ 

 

= 15 + 16 = 31

Co factor of A₃₂

Co factor of A₃₂   element  can  be calculated   by eliminating 3rd row and 2nd column and solving the remaining determinant.

= (-1)³⁺²M₃₂ 

= -1(10 - 12) = 2

Co factor of A₃₃

Co factor of A₃₃   element  can  be calculated   by eliminating 3rd row and 3rd column and solving the remaining determinant.

= (-1)³⁺³M₃₃ 

= - 8 - 9 = -17

Co Factor Matrix of A

Now write all the co factors calculated in Matrix Form as follows

Ad joint Matrix of A

To find the Ad Joint of  Co factor Matrix, transform 1st row to 1st column, 2nd row to 2nd column  and 3rd rows to 3rd column as follows 

Formula to Find   A^-1 

To Find the value of  A^-1 ,Putting the value of Adj A in the formula given below.

Find values of x ,y and z

Multiplying both the matrices which are on the right side.

Simplifying the matrix so obtained

Dividing each element by 86 to get matrix of 3×1 (i. e. Perform scalar multiplication of matrix ).

Using the equality of two matrices .We can write the values of x, y  and z respectively like this

 x =277/86  , y = 4/86  and z = -77/86

Now the values of P, Q and R can be calculated by putting the values of x , y and z in equations (1) , (2 ) and (3) respectively.

P = 86/277  ,Q = 86/4   and R = -86/77

Don't Forget to Watch this Video of same Method

Verification of solution

Putting values of P , Q and R in given  equations , these values must satisfies  three given equations

From 1st equation

Similarly other two equations can be verified

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Conclusion

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