Determinant

Definition

In mathematics, a determinant is a special number that can be calculated from a matrix. The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, Cramer's rule, calculus, and more.

History

The concept of the determinant was originally introduced by the Japanese mathematician Seki Kōwa in the 17th century. However, the term "determinant" was coined by Gauss in 1801. The development and study of determinants has been a significant topic in the field of linear algebra and beyond.

Calculation

The determinant of a matrix can be calculated by a number of methods, including the Laplace's expansion, cofactor expansion, and the row operations.

Laplace's Expansion

Laplace's expansion, also known as expansion by minors, is a method to compute the determinant of a square matrix. It expresses the determinant as a sum of products of entries and smaller determinants, called minors.

Cofactor Expansion

Cofactor expansion is a method for calculating the determinant of a square matrix. It involves the calculation of several smaller determinants, called minors, and their combination using a specific formula.

Row Operations

Row operations are a set of techniques used in linear algebra to calculate the determinant of a matrix. These operations include row swapping, row multiplication, and row addition.

Properties

Determinants have a number of important properties, including the following:

- The determinant of an identity matrix is 1. - The determinant of a diagonal, triangular, or permutation matrix is the product of the entries of the main diagonal. - The determinant changes its sign when two rows or two columns of the matrix are interchanged. - The determinant is equal to the product of the eigenvalues of the matrix.

Applications

Determinants are used in a wide variety of applications in both pure and applied mathematics. They are essential tools in linear algebra, calculus, and geometry. Some of the key applications of determinants include:

- Solving systems of linear equations using Cramer's rule. - Finding the inverse of a matrix. - Calculating the volume of a parallelepiped in three dimensions. - Determining whether a set of vectors is linearly independent. - Calculating the characteristic polynomial of a matrix, which is used in the study of differential equations.