Question

If a, b and c are non zero 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. $ab + bc + ca$
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Multiplying $\left| \begin{matrix} 1 & 4 & 9 \\ 3 & 5 & 7 \\ 5 & 7 & 9 \end{matrix} \right|$ by $a,R_{2}$ by $b$ and $R_{3}$ by $c,$ we have

ab^{2}c^{2} & abc & ab + ac \\ a^{2}bc^{2} & abc & bc + ab \\ a^{2}b^{2}c & abc & ac + bc \end{matrix} \right|$$ = $\frac{a^{2}b^{2}c^{2}}{abc}\left| \begin{matrix} bc & 1 & ab + ac \\ ac & 1 & bc + ab \\ ab & 1 & ac + bc \end{matrix} \right| = abc\left| \begin{matrix} bc & 1 & \Sigma ab \\ ac & 1 & \Sigma ab \\ ab & 1 & \Sigma ab \end{matrix} \right|$ {by $C_{3} \rightarrow C_{3} + C_{1}$} = $abc.\Sigma ab\left| \begin{matrix} bc & 1 & 1 \\ ca & 1 & 1 \\ ab & 1 & 1 \end{matrix} \right| = 0$, [Since $C_{2} \equiv C_{3}$]. **Trick :** Put $a = 1,b = 2,c = 3$ and check it.