nLab inverse matrix




n×nn\times n matrices with entries in a unital ring RR form a unital ring Mat n×n(R)Mat_{n \times n}(R) with unit I nI_n (diagonal matrix whose each entry at the main diagonal is 1 R1_R). A matrix AMat n×n(R)A\in Mat_{n \times n}(R) is invertible (also said regular) if it has a two-sided inverse A 1A^{-1} in that unital ring, called the matrix inverse (or inverse matrix) of AA. In other words, it is a matrix satisfying AA 1=I n=A 1AA A^{-1} = I_n = A^{-1} A.


One-sided inverses

Sometimes one-sided inverses in Mat n×n(R)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 1A^{-1} satisfying weaker requirement AA 1A=I nA A^{-1} A = I_n and A 1=A 1AA 1A^{-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.



(fundamental theorem of invertible matrices)

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

  1. AA is an invertible matrix;

  2. AA is the matrix product of elementary matrices.


Formulas for inverses of block matrices see shortly at Schur complement and more at

  • 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

  • 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

  • Tzon-Tzer Lu, Sheng-Hua Shiou, Inverses of 2×22 \times 2 block matrices, Computers & Mathematics with Applications 43 1–2 (2002) 119-129 [doi:10.1016/S0898-1221(01)00278-4]

  • Müge Saadetoğlu, Şakir Mehmet Dinsev, Inverses and determinants of n×nn \times n block matrices, Mathematics 11 17 (2023) 3784 [doi:10.3390/math11173784]

Last revised on May 22, 2024 at 18:26:26. See the history of this page for a list of all contributions to it.