Question

If $a,b,c$ are different and $\left| \begin{matrix} a & a^{2} & a^{3} - 1 \\ b & b^{2} & b^{3} - 1 \\ c & c^{2} & c^{3} - 1 \end{matrix} \right| = 0$, then.

• A. $a + b + c = 0$
• B. $abc = 1$
• C. $a + b + c = 1$
• D. $ab + bc + ca = 0$

Answer 1

Answer: (B)

$abc = 1$

Answer 2

$\left| \begin{matrix} a & a^{2} & a^{3} - 1 \\ b & b^{2} & b^{3} - 1 \\ c & c^{2} & c^{3} - 1 \end{matrix} \right| = 0$⇒ $\left| \begin{matrix} a & a^{2} & a^{3} \\ b & b^{2} & b^{3} \\ c & c^{2} & c^{3} \end{matrix} \right| - \left| \begin{matrix} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{matrix} \right| = 0$

⇒ $abc\left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| - \left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| = 0$

⇒ $(abc - 1)\left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| = 0$

Since $a,b,c$ are different, so

Hence $abc - 1 = 0$i.e., $abc = 1$.