$n\times n$ matrices with entries in a unital ring $R$ form a unital ring $Mat_{n \times n}(R)$ with unit $I_n$ (diagonal matrix whose each entry at the main diagonal is $1_R$). A matrix $A\in Mat_{n \times n}(R)$ is invertible (also said regular) if it has a two-sided inverse $A^{-1}$ in that unital ring, called the matrix inverse (or inverse matrix) of $A$. In other words, it is a matrix satisfying $A A^{-1} = I_n = A^{-1} A$.

Variants

One-sided inverses

Sometimes one-sided inverses in $Mat_{n \times n}(R)$ are also useful as well as one-sided inverses of rectangular matrices.

Schur complement

Sometimes, an inverse matrix does not exist but formulas for some entries of inverse matrix make sense. Related expressions include quasideterminants and Schur complement, see there.

Moore–Penrose inverse

A generalized inverse $A^{-1}$ satisfying weaker requirement $A A^{-1} A = I_n$ and $A^{-1} = A^{-1} A A^{-1}$ is sometimes useful, especially in applied mathematics and approximation theory. These identities play role also in inverse semigroups.

In the case of matrices, such generalized inverse is known as Moore–Penrose inverse.

Properties

Proposition

(fundamental theorem of invertible matrices)

If $R = k$ is a field, $n \in \mathbb{N}$ and $A \in Mat_{n \times n}(k)$ a square matrix, the following are equivalent:

D. Krob, B. Leclerc, Sec 2 in: Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995) 1-23 [doi:10.1007/BF02101594, arXiv:hep-th/9411194]đ

Chapter 13 (e.g. 13.8), Block LU factorization, in: Nicholas J. Higham, Accuracy and stability of numerical algorithms, Society for Industrial and Applied Mathematics, Year: 2002