Question

Without expanding, show that the value of the following determinant is zero:

\[\begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]

Answer 1

\[ ∆ = \begin{vmatrix}a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix}\]
\[ = \begin{vmatrix}a + 2x & b + 2y & c + 2z \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix} \left[ \text{ Applying }R_1 \to R_1 + 2 R_3 \right]\]
\[ = \begin{vmatrix}0 & 0 & 0 \\ a + 2x & b + 2y & c + 2z \\ x & y & z\end{vmatrix} = 0 \left[ \text{ Applying }R_1 \to R_1 - R_2 \right]\]