Q. If $D = \left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{matrix} \right|for\ x \neq 0,\ y \neq 0,\$then D is
A) Divisible by neither $x$ nor $y$
B) Divisible by both $x$ and $y$
C) Divisible by $x$ but not $y$
D) Divisible by $y$ but not $x$
Explanation
(b)
Given, $D = \left| \begin{matrix}
1 & 1 & 1 \\
1 & 1 + x & 1 \\
1 & 1 & 1 + y
\end{matrix} \right|$
Applying $C_{2} \rightarrow C_{2} - C\$and$\ C_{3} \rightarrow C_{3} - C_{1}$
$= \left| \begin{array}{r}
1\ \ 0\ \ 0 \\
1\ \ x\ \ 0 \\
1\ \ 0\ \ y
\end{array} \right| = xy$
Hence, $D$ is divisible by both $x$ and $y$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.