Which of the following has/have value equal to zero?

Moderate

A

$|\begin{array}{ccc} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{array}|$
B

$|\begin{array}{lll} 1 / a a^{2} bc \\ 1 / b b^{2} ac \\ 1 / c c^{2} ab \end{array}|$
C

$|\begin{array}{ccc} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{array}|$
D

$|\begin{array}{lll} 2 43 6 \\ 7 35 4 \\ 3 17 2 \end{array}|$

Solution

$\begin{matrix} |\begin{array}{ccc} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{array}| = |\begin{array}{ccc} 8 & 2 & 7 \\ 4 & 1 & - 2 \\ 4 & 1 & - 2 \end{array}| [R_{3} \to R_{3} - R_{2} and R_{2} \to R_{2} - R_{1}] \\ = 0 \end{matrix}$
$\begin{aligned} |\begin{array}{ccc} 1 / a & a^{2} & bc \\ 1 / b & b^{2} & ac \\ 1 / c & c^{2} & ab \end{array}| = \frac{1}{abc} |\begin{array}{ccc} 1 & a^{3} & abc \\ 1 & b^{3} & abc \\ 1 & c^{3} & abc \end{array}| [R_{1} \to {aR}_{1}, R_{2} \to {bR}_{2}, R_{3} \to {cR}_{3}] \end{aligned}$
$= \frac{abc}{abc} |\begin{array}{lll} 1 a^{3} 1 \\ 1 b^{3} 1 \\ 1 c^{3} 1 \end{array}|$ [taking abc common from C₃]
$\begin{matrix} = 0 \\ \begin{aligned} |\begin{array}{ccc} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{array}| & = |\begin{array}{ccc} a + b & 2 a + b & 3 a + b \\ a & a & a \\ 2 a & 2 a & 2 a \end{array}| \\ = 0 \end{aligned} [R_{3} \to R_{3} - R_{2}, R_{2} \to R_{2} - R_{1}] \end{matrix}$
$\begin{matrix} |\begin{array}{lll} 2 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{array}| = |\begin{array}{lll} 2 & 1 & 6 \\ 7 & 7 & 4 \\ 3 & 3 & 2 \end{array}| [C_{2} \to C_{2} - 7 C_{3}] \\ = |\begin{array}{lll} 1 & 1 & 6 \\ 0 & 7 & 4 \\ 0 & 3 & 2 \end{array}| [C_{1} \to C_{1} - C_{2}] \\ = 2 \end{matrix}$

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