Question

The value of the following determinant $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|$ is:

• A. $(a-b)(b-c)(c-a)(a+b+c)$
• B. $abc(a+b)(b+c)(c+a)$
• C. $(a-b)(b-c)(c-a)$
• D. none of the above

Answer 1

Answer: (A)

Let $\Delta =\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|$
Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$
$=\left|\begin{array}{ccc}1& 0& 0\\ a^3& b-a& c-a\\ a^3& b^3-a^3& c^3-a^3\end{array}\right|$
$=(b-a)(c-a)$
$\times \left|\begin{array}{ccc}1& 0& 0\\ a& 1& 1\\ a^3& b^2+ab+a^2& c^2+ac+a^2\end{array}\right|$
$=(b-a)(c-a)\times \left[\left(c^2+ac+a^2\right)-\left(b^2+ab+a^2\right)\right]$
$=(b-a)(c-a)\left[c^2-b^2+ac-ab\right]$
$=(b-a)(c-a)(c-b)[c+b+a]$
Note: The value of the determinant does not change either by any rows or by any columns operation.