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 elementsFind 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
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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