Question

$\left|\begin{array}{ccc}1& 2& 3\\ 1^3& 2^3& 3^3\\ 1^5& 2^5& 3^5\end{array}\right|$ equal to

• A. $1!213$
• B. $1!3!5!$
• C. $6!$
• D. $9!$

Answer 1

Answer: (C)

Let $A=\left|\begin{array}{ccc}1& 2& 3\\ 1^3& 2^3& 3^3\\ 1^5& 2^5& 3^5\end{array}\right|$
$=\mathrm{1.2.3}\left|\begin{array}{ccc}1& 1& 1\\ 1^2& 2^2& 3^2\\ 1^4& 2^4& 3^4\end{array}\right|$
$=6\left|\begin{array}{ccc}1& 1& 1\\ 1& 4& 9\\ 1& 16& 81\end{array}\right|$
$=6\left|\begin{array}{ccc}1& 0& 0\\ 1& 3& 5\\ 1& 15& 65\end{array}\right|(C_2\to C_2-C_1,C_3\to C_3-C_2)$
$=\mathrm{6.3.5}\left|\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 5& 13\end{array}\right|$
$=90[1(13-5)]$
$=90\times 8$
$=720=6!$