# Determinant Formula

Determinant in linear algebra is a useful value which is computed from the elements of a square matrix. The determinant of a matrix A is denoted det (A), det A, or |A|. It is a function that has an input accepts $n\times n$ matrix and output in a real or a complex number which is called as the determinant of the input matrix. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant is used to solve these equations, even though more efficient techniques are actually used, some are determinant-revealing and consist of computationally effective ways of calculating the determinant itself.

If $\LARGE A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

#### The Determinant Formula is

$\LARGE Determinant=ad-bc$

If $\LARGE B=\begin{bmatrix} i & j & k\\ a & b & c\\ x & y & z \end{bmatrix}$

#### The Determinant Formula is

$\LARGE Determinant =i(bz-cy)-j(az-cx)+k(ay-bx)$

### Solved Examples

Question 1: Find out the determinant of $\begin{bmatrix} 4 & 2\\ 7 & 8 \end{bmatrix}$

Solution:

Given 2 $\times$ 2 matrix is,

A = $\begin{bmatrix} 4 & 2\\ 7 & 8 \end{bmatrix}$

Determinant is calculated as,

ad – bc = (4 $\times$ 8) – (2 $\times$ 7) = 32 – 14 = 18