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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|)

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

 Related Calculator: >> Matrix Inverse & Adjoint Calculator. Related Topics: >> Find Inverse of matrix >> Find determinant of A (|A|) >> Find adjoint of A (adj A)