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 \[\;S{\text{ }} = {\text{ }}\left[ {{\text{ }}{s_{ij}}{\text{ }}} \right]\] 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 \[\;f{\text{ }}:{\text{ }}N{\text{ }} \geqslant {\text{ }}O\] is defined by \[f\left( S \right){\text{ }} = {\text{ }}o\] , where \[S \in N\]and\[o \in O\] , then \[f\left( S \right)\] is called the determinant of $S$ . It is also denoted by \[\left| {{\text{ }}S{\text{ }}} \right|\] or \[det{\text{ }}S\] .
If \[S{\text{ }} = \left( {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right)\]
Then the value of determinant of $S$ is given as \[\left| {{\text{ }}S{\text{ }}} \right|{\text{ }} = \] \[\left| {{\text{ }}\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right|\]
\[ = {\text{ }}det{\text{ }}\left( {{\text{ }}S{\text{ }}} \right){\text{ }}.\]
For matrix $S$ , \[\left| {{\text{ }}S{\text{ }}} \right|\] 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 \[S{\text{ }} = {\text{ }}\left[ {{\text{ }}s{\text{ }}} \right]\] be the matrix of order $1$ , then the determinant of $S$ is defined to be equal to $s$ .
2. Order Two :
Let \[S{\text{ }} = \left( {\begin{array}{*{20}{c}}
  {{s_{11}}}&{{s_{12}}} \\
  {{s_{21}}}&{{s_{22}}}
\end{array}} \right)\] be a matrix of order \[2{\text{ }} \times {\text{ }}2\] , then
The determinant of $S$ is defined as :
\[det{\text{ }}\left( {{\text{ }}S{\text{ }}} \right){\text{ }} = {\text{ }}{s_{11}}{\text{ }} \times {\text{ }}{s_{22}}{\text{ }} - {\text{ }}{s_{12}}{\text{ }} \times {\text{ }}{s_{21}}\]
3. Order three :
$\left( {\begin{array}{*{20}{c}}
  {{s_{11}}}&{{s_{12}}}&{{s_{13}}} \\
  {{s_{21}}}&{{s_{22}}}&{{s_{23}}} \\
  {{s_{31}}}&{{s_{32}}}&{{s_{33}}}
\end{array}} \right)$
The determinant of S of the order \[3{\text{ }} \times {\text{ }}3\] is written as :
\[det{\text{ }}\left( {{\text{ }}S{\text{ }}} \right){\text{ }} = {\text{ }}{s_{11}}{\text{ }} \times {\text{ }}\left[ {{\text{ }}{s_{22}}{\text{ }} \times {\text{ }}{s_{33}}{\text{ }} - {\text{ }}{s_{32}} \times {\text{ }}{s_{23}}} \right]{\text{ }} - {\text{ }}{s_{12}}{\text{ }} \times {\text{ }}\left[ {{\text{ }}{s_{21}}{\text{ }} \times {\text{ }}{s_{33}}{\text{ }} - {\text{ }}{s_{31}} \times {\text{ }}{s_{23}}{\text{ }}} \right]{\text{ }} + {\text{ }}{s_{13}}{\text{ }} \times {\text{ }}\left[ {{\text{ }}{s_{21}}{\text{ }} \times {\text{ }}{s_{32}}{\text{ }} - {\text{ }}{s_{31}}{\text{ }} \times {\text{ }}{s_{22}}{\text{ }}} \right]\]

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 .