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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
131
112
234
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
-201
92-3
51-2
d)Forming Cofactor Matrix of A:

Matrix of cofactors
-2 x 10 x -11 x 1
9 x -12 x 1-3 x -1
5 x 11 x -1-2 x 1
    =    
-201
-923
5-1-2
e)Forming Adjoint A:

Adjoint Matrix (Adj A)
-2-95
02-1
13-2
f) Finding the Inverse Matrix of A

Inverse of Matrix (A-1)
A-1 = ajd A / |A| =     1/-1
-2-95
02-1
13-2
A-1     =
29-5
0-21
-1-32