Question

$\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|=$

• A. $2(10!\cdot 12!\cdot 13!)$
• B. $2(10!\cdot 11!\cdot 12!)$
• C. $-2(10!\cdot 11!\cdot 12!)$
• D. $2(10!\cdot 13!)$

Answer 1

The correct option is B $2(10!\cdot 11!\cdot 12!)$

$\Delta =\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|$

$=10!\cdot 11!\cdot 12!\left|\begin{array}{ccc}1& 1& 1\\ 11& 12& 13\\ 11\cdot 12& 12\cdot 13& 13\cdot 14\end{array}\right|$

$C_2\to C_2-C_1$, $C_3\to C_3-C_1$
$\Rightarrow \Delta =10!\cdot 11!\cdot 12!\left|\begin{array}{ccc}1& 0& 0\\ 11& 1& 2\\ 11\cdot 12& 24& 50\end{array}\right|$

$\Rightarrow \Delta =10!\cdot 11!\cdot 12!(50-48)\Rightarrow \Delta =2(10!\cdot 11!\cdot 12!)$