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 a real inverse. Such matrices are non-invertible.

Calculation of the determinant is the common test for invertibility. if |A| != 0, then A is invertible. Conversely, if |A| = 0, then A is non-invertible.

Only square matrices can be invertible. However, a non-square matrix can separably have distinct left inverse and right inverse matrices. Generally, if a < b, then a matrix with shape a by b and rank of a can have a right inverse; a matrix with shape b by a and rank of a can have a left inverse. By contrast, an invertible square matrix has a true inverse that works on either side: AA^-1 = A^-1A = I.

Properties

The core principle of inversions is that a matrix A can be canceled out from a larger system: xAA^-1 = x.

An invertible matrix has only one vector in the null space: the zero vector. If any basis vector of a matrix is a linear transformation of another, then the matrix does not have basis and must be non-invertible.

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

Calculation

Because AA^-1 = I, applying elimination and backwards elimination on A augmented with an identity matrix (I) will create A^-1 in the augmentation.

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

A^-1 is:

┌      ┐
│  7 -3│
│ -2  1│
└      ┘

Determinant and Cofactor Matrix

Given the determinant of A, it can also be simple to compute A^-1 as (1/|A|)C^T. C is the cofactor matrix, where c_{i j} is the cofactor of a_{i j}.

For example, given a 2 x 2 A like:

┌    ┐
│ a b│
│ c d│
└    ┘

The cofactor matrix C is:

┌      ┐
│  d -c│
│ -b  a│
└      ┘

But this must be transposed to C^T:

┌      ┐
│  d -b│
│ -c  a│
└      ┘

And then A^-1 is:

┌                               ┐
│  (1/det A) * d  (1/det A) * -b│
│ (1/det A) * -c   (1/det A) * a│
└                               ┘

The above example fits into this formula. The elimination and backwards elimination prove that the determinant of that A is 1. The more fundamental formula ad - bc expands to 1 * 7 - 2 * 3 which also reveals a determinant of 1. As such, (1/|A|) is trivially 1. So simply plug the given (a, b, c, d) into the transposed cofactor matrix to find the inverse.

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