# JEE Main & Advanced Mathematics Determinants & Matrices Definition

Definition

Category : JEE Main & Advanced

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 ${{C}_{1}},\,\,{{C}_{2}},\,\,{{C}_{3}},.....$

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