Question

Write the value of the determinant 

\[\begin{vmatrix}a & 1 & b + c \\ b & 1 & c + a \\ c & 1 & a + b\end{vmatrix} .\]

 

Answer 1

\[\text{ Let }∆ = \begin{vmatrix}a & 1 & b + c \\ b & 1 & c + a \\ c & 1 & a + b\end{vmatrix} \] 
\[ = \begin{vmatrix} a + b + c & 1 & b + c\\a + b + c & 1 & c + a\\a + b + c & 1 & a + b \end{vmatrix} \left[\text{ Applying }C_1 \to C_1 + C_3 \right]\] 
\[ = a + b + c \begin{vmatrix} 1 & 1& b + c\\1 & 1 & c + a\\1 & 1 & a + b \end{vmatrix}\] 
\[ = \left( a + b + c \right) \times 0\] 
\[ = 0\]