Invertibility

Invertibility is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the identity matrix. The calculation of an inverse matrix is inversion.

Definition

A matrix A is invertible if there is a matrix A^-1 which satisfies AA^-1 = A^-1A = I.

If a matrix cannot be inverted, it is singular and degenerate and 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 m < n, then a matrix with shape m by n and rank of m can have a right inverse; a matrix with shape n by m and rank of m can have a left inverse.

Properties

By definition, AA^-1 = A^-1A = I.

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.

Test with Determinant

The determinant is the most common test for invertibility. If |A| != 0, then A is invertible. If |A| = 0, then A is non-invertible.

Calculation with Elimination

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│
└      ┘

Calculation with Determinants and Cofactor Matrices

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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