Matrix determinant
Definition and Overview
A matrix determinant is a special number that can be calculated from a square 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, calculus and more.
History
The concept of the determinant was originally developed in the context of systems of linear equations. The Chinese text The Nine Chapters on the Mathematical Art, written as early as 150 BC, solves determinants of order up to 3. The development of determinants in Europe started in the 18th century with the work of Leibniz.
Calculation
The determinant of a matrix can be calculated using various methods, including the Laplace expansion, or the more efficient LU decomposition method. For a 2x2 matrix, the determinant can be calculated as the product of the values on the main diagonal minus the product of the off-diagonal values.
Properties
Determinants have several important properties. They are unchanged by transposition, change sign when two rows are swapped, and are multiplied by a scalar when a row is multiplied by that scalar. They also obey a distributive rule when a row is added to another.
Applications
Determinants are used in a wide variety of mathematical applications, including solving systems of linear equations, finding the inverse of a matrix, and calculating the volume of a parallelepiped. They are also used in calculus, physics, and engineering.
