JEE Main & Advanced Mathematics Determinants & Matrices Definition

Let us consider three homogeneous linear equations

\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z=0\],\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z=0\]

and  \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z=0\]

Eliminated \[x,\,\,y,\,\,z\] from above three equations we obtain

\[{{a}_{1}}({{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}})-{{b}_{1}}({{a}_{2}}{{c}_{3}}-{{a}_{3}}{{c}_{2}})+{{c}_{1}}({{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}})=0\]   …..(i)

The L.H.S. of (i) is represented by  \[\left| \,\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|={{a}_{1}}\,\left| \,\begin{matrix}{{b}_{2}} & {{c}_{2}}  \\{{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|-{{b}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|+{{c}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{b}_{2}}  \\{{a}_{3}} & {{b}_{3}}  \\\end{matrix}\, \right|\]

Its contains three rows and three columns, it is called a determinant of third order.

The number of elements in a second order is \[{{2}^{2}}=4\] and the number of elements in a third order determinant is \[{{3}^{2}}=9\].

Rows and columns of a determinant : In a determinant horizontal lines counting from top \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ }{{3}^{rd}},\ldots ..\] respectively known as rows and denoted by \[{{R}_{1}},\,\,{{R}_{2}},\,\,{{R}_{3}},\,\,......\] and vertical lines counting left to right, \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ }{{3}^{rd}},\ldots ..\] respectively known as columns and denoted by \[{{C}_{1}},\,\,{{C}_{2}},\,\,{{C}_{3}},.....\]