Introduction to Determinant

Determinant is a square matrix that associated with a unique number ( or function).

A matrix which is not square, does not possess determinant.

Let

$A=[a_{ij}]=\begin{bmatrix} a_{11} &a_{12}.........a_{1n}\\ a_{21} &a_{22}.........a_{2n}\\ .......&.................\\ .......&.................\\ a_{n1} &a_{n2}.........a_{nn} \end{bmatrix}$
The determinant of A is denoted by any of the following :

$detA \ , \ \begin{vmatrix} A \end{vmatrix} \ , \ |a_{ij}| \ , \ \begin{vmatrix} a_{11} &a_{12}.........a_{1n}\\ a_{21} &a_{22}.........a_{2n}\\ .......&.................\\ .......&.................\\ a_{n1} &a_{n2}.........a_{nn} \end{vmatrix}$

For example, if
$A=\begin{bmatrix} 4& 3 & 6\\ 8& 7& 5 \\ -1&4 & 6 \end{bmatrix}$ $then \ \ \ |A|=\begin{vmatrix} 4 & 3&6 \\ 8& 7& 5\\ -1&4 & 6 \end{vmatrix}$

The determinant of a square matrix of order n is called a determinant of order n .
If the elements of square matrix A are numbers, then |A| is a number and in case, the
elements of A are functions, then |A| is a function.

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Introduction to Determinant

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