Question

What is the Order of Determinant?

Answer 1

Hint: We have to solve this question by stating the definition of order of determinant . We will also give examples of order of determinants . We will also define the value of order of determinants. We will also mention the various operations of determinants which can be applied to the determinant matrix .

Complete step-by-step solution:
To every square matrix $$S=\left[s_{ij}\right]$$ of order $n$ , we can associate a number ( real or complex ) called determinant of the square matrix $S$ , where $s_{(ij)}=(i,j{)}^{(th)}$element of $S$ . This is a thought function which associates each of the square matrices with a unique value ( real or complex ) . If $N$ is the set of square matrices , $O$ is the set of numbers ( real or complex ) and $$f:N⩾O$$ is defined by $$f\left(S\right)=o$$ , where $$S\in N$$and$$o\in O$$ , then $$f\left(S\right)$$ is called the determinant of $S$ . It is also denoted by $$\left|S\right|$$ or $$detS$$ .
If $$S=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$
Then the value of determinant of $S$ is given as $$\left|S\right|=$$ $$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$$
$$=det\left(S\right).$$
For matrix $S$ , $$\left|S\right|$$ is read as a determinant of $S$ and not as a modulus of S .
Determinant exists only for square matrices only .
Types of determinants :
1. Order One :
Let $$S=\left[s\right]$$ be the matrix of order $1$ , then the determinant of $S$ is defined to be equal to $s$ .
2. Order Two :
Let $$S=\left(\begin{array}{cc}{s}_{11}& s_{12}\\ s_{21}& s_{22}\end{array}\right)$$ be a matrix of order $$2\times 2$$ , then
The determinant of $S$ is defined as :
$$det\left(S\right)=s_{11}\times s_{22}-s_{12}\times s_{21}$$
3. Order three :
$\left(\begin{array}{ccc}{s}_{11}& s_{12}& s_{13}\\ s_{21}& s_{22}& s_{23}\\ s_{31}& s_{32}& s_{33}\end{array}\right)$
The determinant of S of the order $$3\times 3$$ is written as :
$$det\left(S\right)=s_{11}\times \left[s_{22}\times s_{33}-s_{32}\times s_{23}\right]-s_{12}\times \left[s_{21}\times s_{33}-s_{31}\times s_{23}\right]+s_{13}\times \left[s_{21}\times s_{32}-s_{31}\times s_{22}\right]$$

Note: Whenever a row or a column is interchanged with the other then the value of the determinant remains unchanged. If any two rows ( or columns ) of a determinant are interchanged , then the sign of the value of the determinant changes. If any two rows ( or columns ) of a determinant are identical ( all the corresponding elements are the same ) , then the value of the determinant is zero .