Question

If $a, b, c$ are three complex numbers such that $a^2+b^2+c^2=0$ and $\Delta=\begin{vmatrix} b^2+c^2 & a b & a c \\ a b & c^2+a^2 & b c \\ a c & b c & a^2+b^2 \end{vmatrix}=k a^2 b^2 c^2,$ then the value of $k$ is

• A. 1
• B. 2
• C. -2
• D. 4

Answer 1

Answer: (D)

Using $a^2+b^2+c^2=0$, we can write $\Delta$ as
$\Delta=\begin{vmatrix}-a^2 & a b & a c \\a b & -b^2 & b c \\a c & b c & -c^2\end{vmatrix}=a b c\begin{vmatrix}-a & a & a \\b & -b & b \\c & c & -c\end{vmatrix}$
[taking $a, b, c$ common from $C_1, C_2, C_3$ respectively]
$=a^2 b^2 c^2\begin{vmatrix}-1 & 1 & 1 \\1 & -1 & 1 \\1 & 1 & -1\end{vmatrix}$
[taking $a, b, c$ common from $R_1, R_2, R_3$ respectively]
$=a^2 b^2 c^2\begin{vmatrix}0 & 2 & 1 \\2 & 0 & 1 \\0 & 0 & -1
\end{vmatrix}=-2 a^2 b^2 c^2\begin{vmatrix}2 & 1 \\0 & -1
\end{vmatrix}=4 a^2 b^2 c^2$
[applying $C_1 \rightarrow C_1+C_3$ and $C_3 \rightarrow C_2+C_3$ ]
Thus, $k=4$.