Question
If \[D = \left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,1 + x\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,1 + y
\end{array} \right|\] $${\text{for }}x \ne 0,y \ne 0,$$ then $$D$$ is
A.
divisible by $$x$$ but not $$y$$
B.
divisible by $$y$$ but not $$x$$
C.
divisible by neither $$x$$ nor $$y$$
D.
divisible by both $$x$$ and $$y$$
Answer :
divisible by both $$x$$ and $$y$$
Solution :
\[{\rm{Given }}\,\,D = \left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,1 + x\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,1 + y
\end{array} \right|\]
$${\text{Apply }}{R_2} \to {R_2} - {R_1}\,{\text{and }}R \to {R_3} - {R_1}$$
\[\therefore \,\,D = \left| \begin{array}{l}
1\,\,\,\,\,\,\,1\,\,\,\,\,\,\,1\\
0\,\,\,\,\,\,x\,\,\,\,\,\,0\\
0\,\,\,\,\,\,0\,\,\,\,\,\,y
\end{array} \right| = xy\]
Hence, $$D$$ is divisible by both $$x$$ and $$y$$