Question

If $a,b,c$are unequal what is the condition that the value of the following determinant is zero$\Delta = \left| \begin{matrix} a & a^{2} & a^{3} + 1 \\ b & b^{2} & b^{3} + 1 \\ c & c^{2} & c^{3} + 1 \end{matrix} \right|$.

• A. $1 + abc = 0$
• B. $a + b + c + 1 = 0$
• C. $(a - b)(b - c)(c - a) = 0$
• D. None of these

Answer 1

Answer: (A)

$1 + abc = 0$

Answer 2

Splitting the determinant into two determinants, we get $\Delta = \left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| + abc\left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right| = 0$

= $(1 + abc)\lbrack(a - b)(b - c)(c - a)\rbrack = 0$

Because a, b, c are different, the second factor cannot be zero. Hence, option (1), $1 + abc =$0, is correct.