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

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