## 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

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**How to Find A**^{-1}

^{-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.

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Find Co factors of A_{11 }element and write it in C_{11 }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

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Find Co factors of A_{12 }element and write it in C_{12 }position

Here co factor will be calculated by cross multiplication of purple coloured four elements##
Find Co factors of A_{13 }element and write it in C_{13 }position

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

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Find Co factors of A_{21 }element and write it in C_{21 }position

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Find Co factors of A_{22 }element and write it in C_{22 }position

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Find Co factors of A_{23 }element and write it in C_{23 }position

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Find Co factors of A_{31 }element and write it in C_{31 }position

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Find Co factors of A_{31 }element and write it in C_{31 }position

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Find Co factors of A_{31 }element and write it in C_{31 }position

## How to find values of x , y and z

x = 52/22

y = -18/11 and

z = -15/11

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**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.

## My previous Posts

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**Do not Forget to watch this video of same Problem**

You can clear your doubts if any after watching this video

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**Do not Forget to watch this video of same Problem**

You can clear your doubts if any after watching this video

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Conclusion

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