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

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:

d) Forming Cofactor Matrix of A:

e) Forming Adjoint A:

f) Finding the Inverse Matrix of A