Explanation

(c)
Let $A = \left| \begin{matrix}
1 & 2 & 3 \\
1^{3} & 2^{3} & 3^{3} \\
1^{5} & 2^{5} & 3^{5}
\end{matrix} \right| = 1.2.3\left| \begin{matrix}
1 & 2 & 3 \\
1^{2} & 2^{2} & 3^{2} \\
1^{4} & 2^{4} & 3^{4}
\end{matrix} \right|$
$= 6\left| \begin{matrix}
1 & 1 & 1 \\
1 & 4 & 9 \\
1 & 16 & 81
\end{matrix} \right| = 6\left| \begin{matrix}
1 & 0 & 0 \\
1 & 3 & 5 \\
1 & 15 & 65
\end{matrix} \right|$
[$C_{2} \rightarrow C_{2} - C_{1},\ C_{3} \rightarrow C_{3} - C_{2}\rbrack$
$= 6.3.5\left| \begin{matrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 5 & 13
\end{matrix} \right| = 90\left\lbrack 1(13 - 5) \right\rbrack = 720 = 6!$