Question

If $a, b,$ and $c$ are nonzero real numbers, then
$\Delta=\left|\begin{matrix}b^{2}c^{2}&bc&b +c\\ c^{2}a^{2}&ca&c +a\\ a^{2}b^{2}&ab& a +b\end{matrix}\right|$is equal to

• A. $abc$
• B. $a^{2} b^{2} c^{2}$
• C. $bc +ca +ab$
• D. none of these

Answer 1

Answer: (D)

Applying $R_1 \to aR_1, R_2 \to bR_2$ and $R_3 \to cR_3$, we get
$\Delta=\frac{1}{abc}\left|\begin{matrix}ab^{2}c^{2}&a bc &ab +ac\\ a^{2}bc^{2}&a bc &bc +ab\\ a^{2}b^{2}c &a bc &ac+ bc\end{matrix}\right|=\frac{a^{2} b^{2}c^{2}}{abc}\left|\begin{matrix}bc&1&ab+ac\\ ac&1&bc+ab\\ ab&1&ac+bc\end{matrix}\right|$
Applying $C_{3}\to C_{3} + C_{1}$ and taking $\left(bc + ca + ab\right)$ common, we get
$\Delta=a bc\left(bc + ca +ab\right)\left|\begin{matrix}bc&1&1\\ ac&1&1\\ ab&1&1\end{matrix}\right|=0$
$\left[\because C_{2} \,\text{and} \,C_{3}\text{are identical}\right]$