nLab inverse matrix

Properties

Proposition

(fundamental theorem of invertible matrices)

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

1. $A$ is an invertible matrix;

2. $A$ is the matrix product of elementary matrices?.

Related concepts

• rank of a matrix

• matrix group

• general linear group

• matrix equivalence

References

• Wikipedia, Invertible matrix

• WolframMathWorld, Invertible Matrix Theorem

Formulas for inverses of block matrices:

• Tzon-Tzer Lu, Sheng-Hua Shiou, Inverses of $2 \times 2$ 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 $n \times n$ Block Matrices, Mathematics 11 17 (2023) 3784 [doi:10.3390/math11173784]