Necessary and sufficient conditions for a to be invertible: |A| is not equal to 0, r(A)=n, the column (row) vector group of A is linearly independent, and A can be decomposed into the product of several elementary matrices. In addition, if A is an invertible matrix, the inverse matrix of A is unique.

The matrix is introduced as follows:

Matrix, a mathematical term. In mathematics, a matrix is a group of complex numbers or real numbers arranged in a rectangular array, which originated from a square matrix composed of coefficients and constants of equations. This concept was first put forward by British mathematician Kelly in19th century.

Matrix is a common tool in applied mathematics such as advanced algebra and statistical analysis. In physics, matrices have applications in circuit science, mechanics, optics and quantum physics. In computer science, three-dimensional animation also needs matrix. Matrix operation is an important problem in the field of numerical analysis.

Decomposition of a matrix into a combination of simple matrices can simplify the operation of the matrix in theory and practical application. For some widely used and special matrices, such as sparse matrix and quasi-diagonal matrix, there are concrete fast operation algorithms.

For the development and application of matrix related theory, please refer to matrix theory. Infinite-dimensional matrices will also appear in astrophysics, quantum mechanics and other fields, which is the generalization of matrices.

The main branch of numerical analysis is devoted to the development of effective algorithms for matrix calculation, which has been a topic for a century and is an expanding research field. The matrix decomposition method simplifies the theoretical and practical calculation.

The customized algorithm for specific matrix structures (such as sparse matrix and near-angle matrix) speeds up the operation speed in finite element method and other calculations. Infinite matrix appears in planetary theory and atomic theory. A simple example of infinite matrix is the matrix representing the derivative operator of Taylor series of functions.

The concept of matrix was gradually formed in19th century. 1800s, Gauss and William Jordan established Gauss-Jordan elimination method. 1844, German mathematician F. Ferdinand F. Eisenstein discussed "transformation" (matrix) and its product.

1850, the English mathematician james joseph sylvester first used the word matrix.