# Inverse of Matrix - Tutorial

## Inverse of Matrix - Tutorial

##### Inverse of Matrix :

After calculating determinant, adjoint from the matrix as in the previous tutorials a) Find determinant of A (|A|) b) Find adjoint of A (adj A) we will be calculating the inverse using determinant and adjoint c) Calculate the inverse using the formulae A-1 = adjoint A / |A|

##### An Example:

For an example we will find the inverse for the following matrix

Matrix A
 1 3 1 1 1 2 2 3 4

a) Finding determinant of A: |A| = 1x(1x4-3x2) - 3x(1x4-2x2) + 1x(1x3-2x1) |A| = 1x(4-6) - 3x(4-4) + 1x(3-2) = -2+0+1 |A| = -1 b) Finding Minors of A: M11 = 1x4-3x2 = 4-6 = -2 M12 = 1x4-2x2 = 4-4 = 0 M13 = 1x3-2x1 = 3-2 = 1 M21 = 3x4-3x1 = 12-3 = 9 M22 = 1x4-2x1 = 4-2 = 2 M23 = 1x3-2x3 = 3-6 = -3 M31 = 3x2-1x1 = 6-1 = 5 M32 = 1x2-1x1 = 2-1 = 1 M33 = 1x1-1x3 = 1-3 = -2 c) Forming Minors Matrix of A:

Matrix of minors
 -2 0 1 9 2 -3 5 1 -2

d) Forming Cofactor Matrix of A:

Matrix of cofactors
 -2 x 1 0 x -1 1 x 1 9 x -1 2 x 1 -3 x -1 5 x 1 1 x -1 -2 x 1
=
 -2 0 1 -9 2 3 5 -1 -2

 -2 -9 5 0 2 -1 1 3 -2

f) Finding the Inverse Matrix of A

Inverse of Matrix (A-1)
A-1 = ajd A / |A| = 1/-1
 -2 -9 5 0 2 -1 1 3 -2
A-1 =
 2 9 -5 0 -2 1 -1 -3 2