Question

history of methods for finding the inverse matrix

Answer 1

In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss, but it was not invented by him.

Answer 2

Chapter 1
Introduction
System of linear equations hold a special place in computational engineering, chemistry, and physics. In reality, multiple computational problems in technology and science are expressed mathematically as a linear system. Solving computational mathematics problems through using partial differential equations (PDE), integral equations, and boundary value problems in ordinary differential equations (ODE) ,produces the system of linear equations.

The idea for finding the solution to Ax = b can be written as x = A-1 b, where A-1 is the inverse matrix of A. However, in most practical computational problems, it is not recommended to compute the inverse matrix to solve a system of linear equations. There are two classes of methods for finding the inverse matrices. In direct methods, a finite number of arithmetic operations leads to an ”exact” (within round-off errors) solution. Examples of such direct methods include Gauss elimination, Gauss-Jordan elimination, the matrix inverse method, LU factorization and Cholesky factorization.[24].

Methods of the second type are called indirect iterative methods. Iterative methods start with an arbitrary first approximation to the unknown solution These methods are used for finding the inverse matrices which large systems of equations. Example for it Newton's methods for the invers. [22].[23].[24].

Cramer presented his determinant-based formula for solving systems of linear equations (today known as Cramer's Rule) in 1750 [4],[5],[2]. Cramer's Rule is highly inefficient method for finding the inverse of matrices.
Gauss first started to describe matrix multiplication (which he thinks of as an organization of numbers, so he had not yet reached the concept of matrix algebra) and the inverse of the matrix in the particular context of the collection of coefficients of quadratic forms. Gauss developed Gaussian elimination around 1800. Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix. The method is named after Carl Friedrich Gauss, but it was not invented by him [32]
Arthur Cayley (1821–1895) He also showed how to find the inverse of a matrix by the cofactor method. In 1853 he published a note giving, for the first time, the inverse of a matrix. Cayley defined the matrix algebraically using addition, multiplication, scalar multiplication and inverses. He gave a precise explanation of an inverse of a matrix. After using addition, multiplication and inverses with matrices. [31]
LU decomposition (also called LU factorization) factorizes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician. Alan Turing and von Neumann were the 20th century giants in the development of stored-program computers. Turing introduced the LU decomposition of a matrix in 1948. The L

Answer 3

I'm working Master in Mathematics and I need more information about the history of the ways to find inverse matrices and their evolution