Question

The value of the dterminant $\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|$ ,is

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

Answer 1

The correct option is C $2(10!11!12!)$

$\left|\begin{array}{ccc}10!& 11!& 12!\\ 11!& 12!& 13!\\ 12!& 13!& 14!\end{array}\right|$
$=\left|\begin{array}{ccc}10!& 11(10!)& 132(10!)\\ 11!& 12(11!)& 156(11!)\\ 12!& 13(12!)& 182(12!)\end{array}\right|$
$=(10!11!12!)\left|\begin{array}{ccc}1& 11& 132\\ 1& 12& 156\\ 1& 13& 182\end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$=(10!11!12!)\left|\begin{array}{ccc}0& -1& -24\\ 0& -1& -26\\ 1& 13& 182\end{array}\right|$
$=(10!11!12!)\left|\begin{array}{ccc}0& 1& 24\\ 0& 1& 26\\ 1& 13& 182\end{array}\right|$
$=2(10!11!12!)$