माना \( \Delta = \begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \\ \end{vmatrix} \)
संक्रिया \( C_{1} \rightarrow C_{1} + {\color{Cyan} C_{2}} + {\color{Orange} C_{3}} \) से
\[ \Delta = \begin{vmatrix} a+b+2c & {\color{Cyan} a } & {\color{Orange} b } \\ c & {\color{Cyan} b+c+2a } & {\color{Orange} b } \\ c & {\color{Cyan} a } & {\color{Orange} c+a+2b } \\ \end{vmatrix} \] \[ \Delta = \begin{vmatrix} a+b+2c+{\color{Cyan} a }+{\color{Orange} b } & {\color{Cyan} a } & {\color{Orange} b } \\ c+{\color{Cyan} b+c+2a }+{\color{Orange} b } & {\color{Cyan} b+c+2a } & {\color{Orange} b } \\ c + {\color{Cyan} a } + {\color{Orange} c+a+2b } & {\color{Cyan} a } & {\color{Orange} c+a+2b } \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} a+b+2c+a+b & a & b \\ c+b+c+2a+b & b+c+2a & b \\ c+a+c+a+2b & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} {\color{Cyan} a}+{\color{Orange} b}+2c+{\color{Cyan} a}+b & a & b \\ {\color{Magenta} c }+{\color{Orange} b}+{\color{Magenta} c }+2a+{\color{Orange} b} & b+c+2a & b \\ {\color{Magenta} c }+{\color{Cyan} a}+{\color{Magenta} c } +{\color{Cyan} a}+2b & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} {\color{Cyan} 2a}+{\color{Orange} 2b}+2c & a & b \\ 2a + {\color{Orange} 2b} + {\color{Magenta} 2c } & b+c+2a & b \\ {\color{Cyan} 2a}+ 2b +{\color{Magenta} 2c } & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} 2 \left ( {\color{Cyan} a}+{\color{Orange} b}+ c \right ) & a & b \\ 2 \left ( a + {\color{Orange} b} + {\color{Magenta} c } \right ) & b+c+2a & b \\ 2 \left ( {\color{Cyan} a}+ b +{\color{Magenta} c } \right ) & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} 2\left ( a+b+c \right ) & a & b \\ 2\left ( a+b+c \right ) & b+c+2a & b \\ 2\left ( a+b+c \right ) & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \begin{vmatrix} {\color{Cyan} 2 \left ( a+b+c \right )} & a & b \\ {\color{Cyan} 2 \left ( a+b+c \right )} & b+c+2a & b \\ {\color{Cyan} 2 \left ( a+b+c \right )} & a & c+a+2b \\ \end{vmatrix} \]
\( {\color{Cyan} C_{1}} \) से \( {\color{Cyan} 2 \left ( a+c+c \right )} \) उभयनिष्ठ (common) लेने पर,
\[ \Delta = {\color{Cyan} 2\left ( a+b+c \right )} \begin{vmatrix} {\color{Cyan} 1} & a & b \\ {\color{Cyan} 1} & b+c+2a & b \\ {\color{Cyan} 1} & a & c+a+2b \\ \end{vmatrix} \]
\[ \Delta = \left ( a+b+c \right ) \begin{vmatrix} 1 & a & b \\ 1 & b+c+2a & b \\ 1 & a & c+a+2b \\ \end{vmatrix} \]
संक्रिया \( {\color{Orange} R_{2}} \rightarrow {\color{Orange} R_{2}} - {\color{Cyan} R_{1}} \) और \( {\color{Magenta} }{\color{Magenta} R_{3}} \rightarrow {\color{Magenta} R_{3}} - {\color{Cyan} R_{1}} \) से
\[ \Delta = \begin{vmatrix} {\color{Cyan} 1} & {\color{Cyan} a} & {\color{Cyan} b} \\ {\color{Orange} 1}-{\color{Cyan} 1} & {\color{Orange} b+c+2a} - {\color{Cyan} a} & {\color{Orange} b} -{\color{Cyan} b} \\ {\color{Magenta} 1} -{\color{Cyan} 1} & {\color{Magenta} a}-{\color{Cyan} a} & {\color{Magenta} c+a+2b}- {\color{Cyan} b} \\ \end{vmatrix} \]
\[ \Delta = \left ( a+b+c \right ) \begin{vmatrix} 1 & a & b \\ 1-1 & b+c+2a-a & b-b \\ 1-1 & a-a & c+a+2b-b \\ \end{vmatrix} \]
\[ \Delta = \left ( a+b+c \right ) \begin{vmatrix} 1 & a & b \\ 0 & a+b+c & 0 \\ 0 & 0 & a+b+c \\ \end{vmatrix} \]
\[ \Delta = \left ( a+b+c \right ) \begin{vmatrix} {\color{Cyan} 1} & {\color{Cyan} a} & {\color{Cyan} b} \\ {\color{Orange} 0} & {\color{Orange} a+b+c} & {\color{Orange} 0} \\ {\color{Magenta} 0} & {\color{Magenta} 0} & {\color{Magenta} a+b+c} \\ \end{vmatrix} \]
\( C_{1} \) के सापेक्ष विस्तार करने पर,
\[ \Delta = {\color{Cyan} 1}\begin{vmatrix} {\color{Orange} a+b+c} & {\color{Orange} 0} \\ {\color{Magenta} 0} & {\color{Magenta} a+b+c} \\ \end{vmatrix} - {\color{Orange} 0} \begin{vmatrix} {\color{Cyan} a} & {\color{Cyan} b} \\ {\color{Magenta} 0} & {\color{Magenta} a+b+c} \\ \end{vmatrix} + {\color{Magenta} 0} + \begin{vmatrix} {\color{Cyan} a} & {\color{Cyan} b} \\ {\color{Orange} a+b+c} & {\color{Orange} 0} \\ \end{vmatrix} \]
\[ \Delta = {\color{Cyan} 1} \left [ {\color{Orange} \left ( a+b+c \right )}.{\color{Magenta} \left ( a+b+c \right )} - {\color{Orange} 0}.{\color{Magenta} 0} \right ] - {\color{Orange} 0} + {\color{Magenta} 0} \]
\[ \Delta = {\color{Orange} \left ( a+b+c \right )}{\color{Magenta} \left ( a+b+c \right )} \]
\[ \Delta = \left ( a+b+c \right ) . \left ( a+b+c \right ) . \left ( a+b+c \right ) \]
\[ \Delta = \left ( a+b+c \right )^{3} \]