Q. The value of the determinant $\left| \begin{matrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{matrix} \right|$ is
A) $2\ (10!11!)$
B) $2\ (10!13!)$
C) $2\ (10!11!12!)$
D) $2\ (11!\ 12!\ 13!)$
Explanation
(c)
Let $\mathrm{\Delta} =$ $\left| \begin{matrix}
10! & 11! & 12! \\
11! & 12! & 13! \\
12! & 13! & 14!
\end{matrix} \right|$
$= (10!)(11!)(12!)\ \left| \begin{matrix}
1 & 11 & 11 \times 12 \\
1 & 12 & 12 \times 13 \\
1 & 13 & 13 \times 14
\end{matrix} \right|$
$= (10!)(11!)(12!)\left| \begin{matrix}
1 & 11 & 11 \times 12 \\
0 & 1 & 24 \\
0 & 2 & 50
\end{matrix} \right| = 2(10!)(11!)(12!)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.