Question

$\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|=3abc$

Answer 1

$$\begin{gathered} ∆=\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right| \\ =\left|\begin{array}{ccc}b+c& a& a\\ b+c& 2c+a& a+2b\\ c& c& a+b\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_2+R_3\right] \\ =2\left|\begin{array}{ccc}b+c& a& a\\ 0& c& b\\ c& c& a+b\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_1-R_2\mathrm{and}\mathrm{taking}\mathrm{out}2\mathrm{common}\mathrm{from}{R}_2\right] \\ =2\left|\begin{array}{ccc}b+c& a& a\\ 0& c& b\\ c& 0& a\end{array}\right|\left[\mathrm{Applying}{R}_3\to R_2-R_3\right] \\ =2\left|\begin{array}{ccc}b& a& 0\\ 0& c& b\\ c& 0& a\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1-R_3\right] \\ =4abc\left[\mathrm{Expanding}\right] \end{gathered}$$