In this page inverse method 3x3 matrix we are going to see how to solve the given linear equation using inversion method.
Formula:
This is the formula that we are going to use to solve any linear equations.
X = A^{1} B
Example 1 :
Solve the following linear equation by inversion method
2x  y + 3z = 9
x + y + z = 6
x  y + z = 2
Solution :
First we have to write the given equation in the form AX = B. Here X represents the unknown variables. A represent coefficient of the variables and B represents constants.inverse method 3x3 matrix



To solve this, we have to apply the formula X = A^{1}B
A 

= 2 

+1 

+3 

= 2 [1 + 1] + 1 [1  1] + 3 [1  1]
= 2 [2] + 1 [0] + 3 [2]
= 4 + 0 6
=  2 ≠ 0
Since A is a non singular matrix. A^{1} exists.
minor of 2 


= 1  (1) = 1 + 1 = 2 
minor of 1 


= 1  1 = 0 
minor of 3 


= 1  1 = 2 
minor of 1 


inverse method 3x3 matrix 
= 1  (3) = 1 + 3 = 2 
minor of 1 


inverse method 3x3 matrix 
= 2  3 = 1 
minor of 1 


inverse method 3x3 matrix 
= 2  (1) = 2 + 1 = 1 
minor of 1 


= 1  3 = 4 inversion method in3x3 matrices 
minor of 1 


= 2  3 = 1 
minor of 1 


= 2  (1) = 2 + 1 = 3 
minor matrix = 


cofactor matrix = 


Adj A = 

A^{1 }= 1/2 

X = A^{1}B
= 

= 

x 


x 


x 


 

 


So, the solutions are x = 1, y = 2 and z = 3
Practice Questions :  
1) Solve the following homogeneous system of linear equations using inversion method 
2x + y + z = 5 x + y + z = 4 x  y + 2z = 1 
2) Solve the following homogeneous system of linear equations using inversion method 
x + 2y + z = 7 2x  y + 2z = 4 x + y  2z = 1 
3) Solve the following homogeneous system of linear equations using inversion method 
x + y + z = 4 x  y + z = 2 2x + y  z = 1 
4) Solve the following homogeneous system of linear equations using inversion method 
2x + 5y + 7z = 52 x + y + z = 9 2x + y  z = 0 
5) Solve the following homogeneous system of linear equations using inversion method 
3x + y  z = 2 2x  y + 2z = 6 2x + y  2z = 2 
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