Minor matrix

Writing:

mn=minor(matrix, x, y);

Returns Minor mn from the matrix matrix of x and column y (use numbers, indexes start at 0).

The minor of the matrix is one of the most important mathematical functions used in linear algebra. It is widely used in various fields of science, technology and economics, playing a key role in solving problems related to linear systems of equations, determining the rank of a matrix, finding the inverse matrix and many others. In this article, we will consider the definition of a minor matrix, its properties and the main ways of application.

The minor of the matrix is the determinant of the square submatrix of the original matrix. For example, let's give a matrix A of size n x n, then it has (n-1) x (n-1) minors obtained by crossing out one row and one column from the original matrix. Let's denote the minors as M[i, j], where i and j are the row and column numbers of the crossed out element, respectively.

The minors of the matrix are often used to solve systems of linear equations. For example, to find unknown coefficients in a linear system, you can make a matrix of coefficients for unknowns and find its minors. After that, using Kramer's rule, you can find the values of the unknowns.

Another important application of matrix minors is to determine the rank of the matrix, that is, the maximum number of linearly independent rows or columns. The rank of a matrix is equal to the largest order of nonzero minors of this matrix. Rank determination is one of the key concepts in linear algebra, its application is widespread in the field of research of matrices and their properties.

Another important area of application of matrix minors is finding the inverse matrix. The inverse matrix is a matrix B such that the product of the original matrix A by the inverse matrix B gives the unit matrix. The inverse matrix exists only for those square matrices whose determinant is not zero. To find the inverse matrix, you can use the Gauss-Jordan algorithm, which is based on reducing the expanded matrix to a diagonal form.

In addition, matrix minors are used in solving optimization problems, statistics, graph theory and many other fields of science and technology where data analysis, modeling and forecasting are required.

In conclusion, the minor of the matrix is an important mathematical function with wide application possibilities. It is used to solve linear systems of equations, determine the rank of a matrix, find the inverse matrix, and in many other mathematical and applied problems. Studying the minors of the matrix allows you to understand more deeply the properties and capabilities of matrix operations, as well as apply them in practice to develop effective and reliable algorithms.