Matrix Inversion

Given a matrix A and an inverse matrix A^-1, the product is the identity matrix. This is an important property with several utilities.

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Definition

For some matrices A, the inverse matrix (A^-1) is a matrix which can be multiplied by the original matrix to produce the identity matrix.

Not all matrices have an inverse matrix. If A^-1 exists, then A is invertible and non-singular.

For a permutation matrix P, 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

Consider the below system, which shows an unknown matrix (A^-1) multiplied by a known matrix (A) creating an identity matrix (I).

         -1
  A     A    =   I

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

The inverse matrix is calculated with elimination and reverse elimination. Augment A with I.

The elimination proceeds as:

┌            ┐
│ [1] 3 │ 1 0│
│  2  7 │ 0 1│
└            ┘
┌               ┐
│ [1]  3  │  1 0│
│  0  [1] │ -2 1│
└               ┘

The reverse elimination proceeds as:

┌             ┐
│ 1  3  │  1 0│
│ 0 [1] │ -2 1│
└             ┘
┌                ┐
│ [1]  0  │  7 -3│
│  0  [1] │ -2  1│
└                ┘

A^-1 is:

┌      ┐
│  7 -3│
│ -2  1│
└      ┘