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 $n$-dimensional volumes of $n$-dimensional parallelepipeds. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down.

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 $[A]$ is denoted $\det(A)$, $\det\ A$, or $\left | A \right |$. In the case where the matrix entries are written out in full, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the brackets or parentheses of the matrix.

For instance, the determinant of the matrix $\begin{bmatrix} a & b \\ d & e \end{bmatrix}$ is written $\begin{vmatrix} a & b \\ d & e \end{vmatrix}$.

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 $2 \times 2$ matrix, $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$, 

the determinant $\begin{vmatrix} a & b\\ c & d \end{vmatrix}$ is defined to be $ad-bc$.

Example 1:  Find the determinant of the following matrix: 

$\displaystyle \begin{bmatrix} 4 & -2 \\ 7 & 5 \end{bmatrix}$

The determinant $\begin{vmatrix} 4 & -2\\ 7 & 5 \end{vmatrix}$ is:

 $\displaystyle \begin{aligned} (4 \cdot 5) - (-2 \cdot 7)&= 20 - (-14)\\ &=34 \end{aligned}$