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Question
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
Solution
\[ ∆ = \begin{vmatrix}1 & a & a^2 - bc \\ 1 & b & b^2 - ac \\ 1 & c & c^2 - ab\end{vmatrix}\]
\[ = \begin{vmatrix}0 & a - b & a^2 - bc - b^2 + ac \\ 0 & b - c & b^2 - ac - c^2 + ab \\ 1 & c & c^2 - ab\end{vmatrix} \left[ \text{ Applying } R_1 \to R_1 - R_2 , R_2 \to R_2 - R_3 \right]\]
\[ = \begin{vmatrix}0 & a - b & \left( a - b \right)\left( a + b \right) + c\left( a - b \right) \\ 0 & b - c & \left( b - c \right)\left( b + c \right) + a\left( b - c \right) \\ 1 & c & c^2 - ab\end{vmatrix}\]
\[ = \left( a - b \right)\left( b - c \right)\begin{vmatrix}0 & 1 & \left( a + b + c \right) \\ 0 & 1 & \left( a + b + c \right) \\ 1 & c & c^2 - ab\end{vmatrix}\]
\[ \Rightarrow ∆ = 0\]
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