## 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  + 1  + 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

 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 = 5x + y + z = 4x - y + 2z = 1 2) Solve the following homogeneous system of linear equations using inversion method x + 2y + z = 72x - y + 2z = 4x + y - 2z = -1 3) Solve the following homogeneous system of linear equations using inversion method x + y + z = 4x - y + z = 22x + y - z = 1 4) Solve the following homogeneous system of linear equations using inversion method 2x + 5y + 7z = 52x + y + z = 92x + y - z = 0 5) Solve the following homogeneous system of linear equations using inversion method 3x + y - z = 22x - y + 2z = 62x + y - 2z = -2

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