Inversion Method





In this page inversion method we are going to see how to solve the given linear equations by using this method. This is one of the most familiar method in solving linear equations.

Formula for inversion method

   X = A⁻¹ B

Example 1:

Solve the following linear equation by inversion-method

x + y = 3

2x + 3y = 8

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.

 
1 1
2 3
 
 
 
x
y
 
 
=
 
3
8
 
 

To solve this we have to apply the formula X = A⁻¹ B

|A|

=
 
1 1
2 3
 
 
=
 
1 1

2 3
 
 

    = 3 - 2

    = 1 ≠ 0

Since A is a non singular matrix. A⁻¹ exists.

Adj A

=
 
3 -1

-2 1
 
 

=1/1
 
3 -1

-2 1
 
 

A⁻¹ =

 
3 -1

-2 1
 
 

X =

A⁻¹ B

 =

 
3 -1

-2 1
 
 

 
3

8
 
 
=
 
3 -1
 
x
 
3
8
 
 
 
-2 1
 
x
 
3
8
 


 
x
y
 
 
=
 
(9-8)
(-6-8)
 
 
x

y
 
 
=
 
1

-14
 

Solution:

X = 1

Y = -14


Example 2:

Solve the following linear equation by inversion-method

2x - y = 7

3x - 2y = 11

Solution:

 
2 -1
3 -2
 
 
 
x
y
 
 
=
 
7
11
 
 

To solve this we have to apply the formula X = A⁻¹ B

|A|

=
 
2 -1
3 -2
 
 
=
 
2 -1

3 -2
 
 

    = -4 - (-3)

    = -4 +3

    = -1

    = -1 ≠ 0

Adj A

=
 
-2 1

-3 2
 
 

=1/(-1)
 
-2 1

-3 2
 
 

=-1
 
-2 1

-3 2
 
 

A⁻¹ =

 
2 -1

3 -2
 
 

X =

A⁻¹ B

 =

 
2 -1
3 -2
 
 
 
7
11
 
 
=
 
2 -1
 
x
 
7
11
 
 
 
3 -2
 
x
 
7
11
 
 
x
y
 
 
=
 
(14-11)
(21-22)
 
 
x

y
 
 
=
 
3

-1
 

Solution:

X = 3

Y = -1

Related pages






HTML Comment Box is loading comments...
















































1. Matrix Rank Calculator

2. Matrix Inverse Calculator - 2x2 Matrix

3. Matrix Inverse Calculator - 3x3 Matrix

4. Matrix Inverse Calculator - 4x4 Matrix

5. Cramer's Rule Calculator - 3x3 Matrix

6. Matrix Addition Calculator - 3x3 Matrix

7. Matrix Subtraction Calculator - 3x3 Matrix

8. Matrix Multiplication Calculator - 2x2 Matrix

9. Matrix Multiplication Calculator - 3x3 Matrix

10. Matrix Determinant Calculator - 3x3 & 2x2 Matrix

11. Matrix Addition Calculator - 2x2 Matrix

12. Matrix Subtraction Calculator- 2x2 Matrix

13. Matrix Addition Calculator - 4x4 Matrix

14. Matrix Subtraction Calculator- 4x4 Matrix

15. Matrix Multiplication Calculator - 4x4 Matrix

16. Matrix Determinant Calculator - 4x4 matrix

17. Squared Matrix Calculator

18. Transpose Matrix Calculator