Question

Write the value of the determinant $\left|\begin{array}{ccc}a& 1& b+c\\ b& 1& c+a\\ c& 1& a+b\end{array}\right|.$

Answer 1

$$\begin{gathered} \mathrm{Let}∆=\left|a1b+c \\ b1c+a \\ c1a+b\right| \\ =\left|a+b+c1b+c \\ a+b+c1c+a \\ a+b+c1a+b\right|\left[\mathrm{Applying}{C}_1\to C_1+C_3\right] \\ =a+b+c\left|11b+c \\ 11c+a \\ 11a+b\right| \\ =\left(a+b+c\right)\times 0 \\ =0 \end{gathered}$$