INVERSE METHOD 3X3 MATRIX

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

 
2 -1 3
1 1 1
1 -1 1
 
 
 
x
y
z
 
 
=
 
9
6
2
 
 

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

|A|

=
 
2 -1 3
1 1 1
1 -1 1
 
 

 = 2

 
1 1

-1 1
 

+1

 
1 1

1 1
 

+3

 
1 1

1 -1
 

  = 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
 

 = 1 - (-1)

 = 1 + 1

 = 2

minor of -1

 
1 1

1 1
 

 = 1 - 1

 = 0

minor of 3

 
1 1

1 -1
 

 = -1 - 1

 = -2

minor of 1

 
-1 3

-1 1
 

inverse method 3x3 matrix

 = -1 - (-3)

 = -1 + 3

 =  2

minor of 1

 
2 3

1 1
 

inverse method 3x3 matrix

 = 2 - 3

 = -1 

minor of 1

 
2 -1

1 -1
 

inverse method 3x3 matrix

 = -2 - (-1)

 = -2 + 1

 = -1

minor of 1

 
-1 3

-1 1
 

 = -1 - 3

 = -4             inversion method in3x3 matrices

minor of -1

 
2 3

1 1
 

 = 2 - 3

 = -1

minor of 1

 
2 -1

1 1
 

 = 2 - (-1)

 = 2 + 1

 = 3

minor matrix =

 
2 0 -2
2 -1 -1
-4 -1 3
 

cofactor matrix =

 
2 0 -2
-2 -1 1
-4 1 3
 

Adj A =

 
2 -2 -4
0 -1 1
-2 1 3
 

A-1 = 1/2

 
-2 2 4
0 1 -1
2 -1 -3
 

X  =  A-1B

  =

 
-2 2 4
0 1 -1
2 -1 -3
 
 
x
y
z
 
 
=
 
-2 2 4
 
x
 
9
6
2
 
 
 
0 1 -1
 
x
 
9
6
2
 
 
 
2 -1 -3
 
x
 
9
6
2
 
 
x
y
z
 
 
=1/2
 
(-18+12+8)
(0+6-2)
(18-6-6)
 
 
x
y
z
 
 
=1/2
 
2
4
6
 
 
x
y
z
 
 
=
 
1
2
3
 

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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