What is a determinant?
A matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. In analytical geometry, determinants express the signed
It can be proven that any matrix has a unique inverse if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.
The determinant of a matrix
For instance, the determinant of the matrix
Determinant of a 2-by-2 Matrix
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, shown below:
For a
the determinant
Example 1: Find the determinant of the following matrix:
$\displaystyle
\begin{bmatrix} 4 & -2 \\ 7 & 5 \end{bmatrix}$
The determinant