Matrix Inversion

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. The calculation of an inverse matrix, if it exists, is called inversion.

Definition

An inverse matrix satisfies the equation AA^-1 = I.

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

Properties

The core principle of inversions is that a matrix A can be canceled out from a larger equation. AA^-1 = I, so the two terms cancel out.

An invertible matrix has only one vector in the null space: the zero vector.

For orthogonal matrices (such as permutation matrices), the inverse is also the transpose: Q^-1 = Q^T.

For a square matrix A, the left inverse is the same as the right inverse. AA^-1 = A^-1A = I

Calculation

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│
└      ┘

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