Inverse Matrices

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.

Consider the below problem:

┌    ┐┌    ┐   ┌    ┐
│ 1 3││ a b│   │ 1 0│
│ 2 7││ c d│ = │ 0 1│
└    ┘└    ┘   └    ┘

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^-1A = I

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│
└      ┘

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