# Inverse matrix calculator

Electrical engineering, physics, chemistry, economics, marketing, psychology - in all these and many other areas of science they use matrices - mathematical objects in the form of rectangular tables.

Basically, a matrix, denoted as A, is a collection of rows and columns at the intersection of which its elements are located. These can be integer, real or complex numbers that are subject to addition, subtraction, multiplication, division, exponentiation, transposition, and other mathematical operations when working with two or more matrices. They can also be represented as functions of two discrete arguments: vertical and horizontal.

Most often, matrices are used for complex recording and solving systems of linear algebraic (and differential) equations. In this case, the number of columns corresponds to the number of unknowns, and the number of rows corresponds to the number of equations.

There is also the concept of an inverse matrix - A-1, which, when multiplied by a regular matrix A, forms a unit matrix - E. The latter is a table whose main diagonal elements are equal to the field unit, and all other elements are equal to zero.

## History of origin and development

Before the concept of “inverse matrix” (A-1) appeared, ancient Chinese, Indian, Arab, and then European scientists studied the ordinary matrix (A). And although today it is an integral element of many applied sciences, just 250-300 years ago matrices were not so common and were used only for highly specialized calculations and mathematical research. Although the first prerequisites for the appearance of matrices arose long before our era.

So, the ancient Chinese magic square, which was essentially a 3x3 square matrix, was known back in 2200 BC. Similar tables were found on the territory of modern India and Arab countries, and clearly indicate that they were used not just for listing/recording number series, but for mathematical calculations, at least for addition and subtraction.

Matrixes became most popular in the middle of the 17th century, as the mathematical theory of determinants developed. One of the founders of the new approach was the Swiss mathematician Gabriel Cramer, who in 1751 proposed a new method for solving systems of linear algebraic equations (SLAEs). Cramer's method implied the use of matrices with a zero determinant of the system.

In the same historical period - at the turn of the 18th-19th centuries - the German scientist Carl Friedrich Gauss used matrices to solve SLAEs, but according to a different principle - with the sequential elimination of unknowns. The Cramer method and the Gauss method are not mutually exclusive, and formed the basis for all subsequent research on this topic.

In the period from 1805 to 1900, it was actively studied by the Irish mathematician William Hamilton, the English scientist Arthur Cayley, the French mathematician Marie Ennemond Camille Jordan and the German researchers Karl Weierstrass and Ferdinand Georg Frobenius.

The latter published his famous scientific work “On Linear Substitutions and Algebraic Equations” in 1878, which involved canonical matrices: square and with two unit diagonals (all other elements in them are equal to zero). After the death of Frobenius - in 1903 - the article “On the Theory of Determinants” was published, which emphasized the axiomatic definition of matrix determinants.

As for the inverse matrix, its research was primarily carried out by Carl Friedrich Gauss and Marie Enmon Camille Jordan. The Gauss-Jordan method implied a row-by-row transformation of the standard matrix A1 and the unit matrix E - until the second one became equal to the inverse - A-1. Today this method is widely used in linear algebra and calculus, and can be used to find inverse transformations and solve linear differential equations.

When talking about the practical application of inverse matrices, we must first mention engineering, optics, cinema and computer games. For example, if you set the angle of rotation of the camera using a matrix, you can use the method of multiplying coordinates to obtain a display of objects on the displays. The coordinate axis in this case will correspond to the product of the camera matrix and the inverse matrix (subject to the rules of their multiplication).

In simple terms, the inverse matrix in digital optics is what the object “lacks” to return to its original position. For example, go back 1 kilometer after a kilometer climb up the mountain. Special effects in movies and the construction of volumetric coordinates in 3D computer games and applications are based on the same principle.