Determinant

A determinant is a scalar value that can be calculated from the elements of a square matrix.

Definition

For every square matrix, $C=[c_{ij}]$ of order $n\times n$ a determinant can be defined as a scalar value that is real or a complex number, where

$c_{ij}$ is the $(i,j)$ th element of matrix $C$ . The determinant can be denoted as $det(C)$ or ${\begin{vmatrix}C\end{vmatrix}}$ .

Determinant is written by taking the grid of numbers and arranging them inside the absolute-value bars instead of using square brackets.

Consider a matrix $C={\begin{bmatrix}4&5\\3&2\end{bmatrix}}$

Then, its determinant can be shown as:

${\begin{vmatrix}C\end{vmatrix}}={\begin{vmatrix}4&5\\3&2\end{vmatrix}}$

Determinant of a matrix of order one

Let $A={\begin{bmatrix}a\end{bmatrix}}$ be the matrix of order $1$ , then determinant of $A$ is defined to be equal to $a$

Example

${\begin{vmatrix}A\end{vmatrix}}={\begin{vmatrix}2\end{vmatrix}}=2$

Determinant of a matrix of order two

Let $A={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}$ be the matrix of order $2\times 2$ , then determinant of $A$ is defined as

${\begin{vmatrix}A\end{vmatrix}}={\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}=a_{11}a_{22}-a_{12}a_{21}$

Example

${\begin{vmatrix}A\end{vmatrix}}={\begin{vmatrix}2&4\\-1&2\end{vmatrix}}=2\times 2-4\times -1=4-(-4)=4+4=8$

Determinant of a matrix of order three

Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column).

$A={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}$

Expansion of a determinant along a row ( First row)

Step 1: Multiply first element $a_{11}$ of first row by $(-1)^{sum\ of\ suffixes\ in\ a_{11}}=(-1)^{1+1}=(-1)^{2}=1$ and with the second order determinant obtained by deleting the elements of first row and first column of ${\begin{vmatrix}A\end{vmatrix}}$ as $a_{11}$ lies in first row and first column.

$1\times a_{11}{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}$

Step 2: Multiply first element $a_{12}$ of first row by $(-1)^{sum\ of\ suffixes\ in\ a_{12}}=(-1)^{1+2}=(-1)^{3}=-1$ and with the second order determinant obtained by deleting the elements of first row and second column of ${\begin{vmatrix}A\end{vmatrix}}$ as $a_{12}$ lies in first row and second column.

$-1\times a_{12}{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}$

Step 3: Multiply first element $a_{13}$ of first row by $(-1)^{sum\ of\ suffixes\ in\ a_{13}}=(-1)^{1+3}=(-1)^{4}=1$ and with the second order determinant obtained by deleting the elements of first row and third column of ${\begin{vmatrix}A\end{vmatrix}}$ as $a_{13}$ lies in first row and third column.

$1\times a_{13}{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}$

Step 4: The expansion of determinant of $A$ ( ${\begin{vmatrix}A\end{vmatrix}}$ ) written as sum of all the three terms obtained in steps 1, 2 and 3 above is given by

${\begin{vmatrix}A\end{vmatrix}}=(1\times a_{11}{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}})+(-1\times a_{12}{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}})+(1\times a_{13}{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}})$

${\begin{vmatrix}A\end{vmatrix}}=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$

Example

$A={\begin{bmatrix}3&1&1\\4&-2&5\\2&8&7\end{bmatrix}}$

${\begin{vmatrix}A\end{vmatrix}}=(1\times 3{\begin{vmatrix}-2&5\\8&7\end{vmatrix}})+(-1\times 1{\begin{vmatrix}4&5\\2&7\end{vmatrix}})+(1\times 1{\begin{vmatrix}4&-2\\2&8\end{vmatrix}})$