Question

The value of the determinant $\left|\begin{array}{ccc}15!& 16!& 17!\\ 16!& 17!& 18!\\ 17!& 18!& 19!\end{array}\right|=$

• A. $15!+16!$
• B. $2\left(15!\right)\left(16!\right)\left(17!\right)$
• C. $15!+16!+17!$
• D. $16!+17!$

Answer 1

The correct option is B

$2\left(15!\right)\left(16!\right)\left(17!\right)$

Explanation for correct option

$$\begin{gathered} \left|\begin{array}{ccc}15!& 16!& 17!\\ 16!& 17!& 18!\\ 17!& 18!& 19!\end{array}\right| \\ =\left(15!\right)\left(16!\right)\left(17!\right)\left|\begin{array}{ccc}1& 16& 16·17\\ 1& 17& 17·18\\ 1& 18& 18·19\end{array}\right| \\ =\left(15!\right)\left(16!\right)\left(17!\right)\left|\begin{array}{ccc}1& 16& 272\\ 1& 17& 306\\ 1& 18& 342\end{array}\right| \\ =\left(15!\right)\left(16!\right)\left(17!\right)\left|\begin{array}{ccc}0& 2& 70\\ 0& 1& 36\\ 1& 18& 342\end{array}\right|\left[R_2\to R_3-R_2,R_1\to R_3-R_1\right] \\ =\left(15!\right)\left(16!\right)\left(17!\right)\left[1\left(2\times 36-1\times 70\right)\right] \\ =\left(15!\right)\left(16!\right)\left(17!\right)\left[72-70\right] \\ =\left(15!\right)\left(16!\right)\left(17!\right)2 \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{B}\right)$