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The determinant of the matrix



Returns the determinant det of the matrix array

The determinant of a matrix is a numeric value that can be obtained from a matrix. The matrix is a rectangular table of numbers divided into rows and columns. To calculate the determinant, it is necessary to apply certain mathematical operations to the elements of the matrix.

The matrix determinant has important applications in various fields of science and engineering, including linear algebra, physics, economics, and computer graphics. Using it allows you to determine whether a matrix is non-degenerate, that is, whether it has an inverse matrix. In addition, the determinant can be used to solve systems of linear equations, find the inverse matrix, calculate the area of a figure or the volume of a parallelepiped.

The formula for calculating the determinant of a matrix depends on its dimension. For a square matrix of dimension n x n, the row or column decomposition method is used. In this case, the matrix decomposes into minors, which are determinants of matrices of smaller dimension. Then there is a recursive calculation of each of the minors and their addition taking into account the sign.

One of the applications of the matrix determinant is the solution of systems of linear equations. If the matrix of the system is non-degenerate, then it has a unique solution that can be expressed using Kramer's formula. Kramer's formula uses the ratio of determinants to express each variable through the determinant of the system and the determinants of the minors formed by replacing a column of free terms with a column of variable values.

In physics, the determinant of the matrix can be used to solve systems of equations describing mechanics or electromagnetism. For example, in hydrodynamics, a determinant can help determine the efficiency of fluid movement through a pipe or channel .

In economics, the determinant of the matrix is used to analyze many financial quantities. For example, a matrix of coefficients in a system of linear equations can represent the relationship between different types of goods and services. The determinant of this matrix indicates the presence or absence of relationships between these goods and services.

In computer graphics, the determinant of a matrix can be used to transform three-dimensional objects such as rotation, scaling, or displacement. It allows you to calculate the new coordinates of the object points after applying the specified transformations.

So, the determinant of the matrix is an important mathematical tool that finds application in various fields. Due to its versatility and wide range of applications, it plays an important role in solving various problems related to linear algebra, physics, economics and computer graphics.

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