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| = Inverse Matrices = | = Matrix Inversion = <<TableOfContents>> ---- |
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If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. ---- == Notable Properties == For a permutation matrix, the inverse is also the transpose: P^-1^ = P^T^. For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I ---- == Calculation == |
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| == Gauss-Jordan Calculation == | === Gauss-Jordan Calculation === |
Matrix Inversion
Introduction
An inverse matrix is a matrix A-1 where multiplying it by matrix A results in the identity matrix.
If A-1 exists, then A is invertible and non-singular. Not all matrices are invertible.
Notable Properties
For a permutation matrix, the inverse is also the transpose: P-1 = PT.
For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I
Calculation
Consider the below problem:
┌ ┐┌ ┐ ┌ ┐ │ 1 3││ a b│ │ 1 0│ │ 2 7││ c d│ = │ 0 1│ └ ┘└ ┘ └ ┘
Gauss-Jordan Calculation
The inverse matrix can be calculated through elimination and reverse elimination.
First step:
┌ ┐
│ [1] 3 │ 1 0│
│ 2 7 │ 0 1│
└ ┘
2 - 1m = 0
m = 2
2 7 0 1
- 1m - 3m - 1m - 0m
____ ____ ____ ____
0 1 -2 1
┌ ┐
│ [1] 3 │ 1 0│
│ 0 1 │ -2 1│
└ ┘Second step:
┌ ┐
│ 1 3 │ 1 0│
│ 0 [1] │ -2 1│
└ ┘
3 - 1m = 0
m = 3
1 3 1 0
- 0m - 1m - -2m - 1m
____ ____ _____ ____
1 0 7 -3
┌ ┐
│ 1 0 │ 7 -3│
│ 0 [1] │ -2 1│
└ ┘The inverse matrix of A is:
┌ ┐ │ 7 -3│ │ -2 1│ └ ┘