The matrix inversion is a matrix operation that can be performed on a square matrix with a determinant that is not 0. The matrix inversion is denoted by a "" operator.

Given the matrix and the inverse

The matrix inversion represents the inverse of the matrix product of and some other value, it's relation to the matrix product can be thought of as the same relation between and

Properties

Given a two invertible matrices and the following properties hold true:

Invertible Matrix Theorem

The invertible matrix theorem states that for any matrix the following statements are always either all true or all false:

is invertible.
has pivots
The columns of are linearly independent
The columns of span
The matrix is invertible
has a unique solution for each in
The rank of is
The determinant of is non-zero
The null space of is 0
0 is not an eigenvalue of

Additionally the linear map formed from also have the same truth value

is invertible
is bijective.

Calculating

We can use Gaussian elimination to determine the inversion of a matrix by placing the matrix on the left side and the identity matrix of the same size on the right side and then use row operations to transform the left side into the identity matrix the resulting right side is the inverted matrix.

Matrix

For a two by two matrix it's inverse can be calculated trivially as follows

Let , then can be defined as