Inverse of a matrix

The inverse of a square matrix A is a second matrix such that AA^-1 = A^-1A = I, I being the identity matrix. There are many ways to compute the inverse, the most common being multiplying the reciprocal of the determinant of A by its adjoint (or adjugate, the transpose of the cofactor matrix). For example,

$A= \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix}$

$A^{-1} = \frac{1}{|A|} \mathrm{adj} \begin{pmatrix} \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} \end{pmatrix} = \frac{1}{|A|} \mathrm{cof} \begin{pmatrix} \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} \end{pmatrix} ^\mathrm{T}$

$= \frac{1}{6-5} \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} ^\mathrm{T} = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}$

This is indeed the inverse of A, as

$\begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

A matrix is invertable if and only if the determinant is not equal to zero.

Pseudoinverse

The inverse of a matrix is normally defined for square matricies. For non-square matrix, a corresponding pseudoinvere matrix can be constructed to produce an identity matrix.