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સાબિત કરો કે $\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|=4 a b c$
Solution
Let $\Delta=\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$
Applying $\quad \mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}-\mathrm{R}_{3}$ to $\Delta,$ we get
$\Delta=\left|\begin{array}{ccc}
0 & -2 c & -2 b \\
b & c+a & b \\
c & c & a+b
\end{array}\right|$
Expanding along $\mathrm{R}_{1},$ we obtain
$\Delta = 0\left| {\begin{array}{*{20}{c}}
{c + a}&b \\
c&{a + b}
\end{array}} \right| – ( – 2c)\left| {\begin{array}{*{20}{c}}
b&b \\
c&{a + b}
\end{array}} \right| + ( – 2b)\left| {\begin{array}{*{20}{c}}
b&{c + a} \\
c&c
\end{array}} \right|$
$ = 2c\left( {ab + {b^2} – bc} \right) – 2b\left( {bc – {c^2} – ac} \right)$
$ = 2abc + 2c{b^2} – 2b{c^2} – 2{b^2}c + 2b{c^2} + 2abc$
$ = 4abc$
