Question

If $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$ for $x\ne 0,y\ne 0$ 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 $x$

Answer 1

Answer: (B)

divisible by both $x$ and $y$

$D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$
Applying $C_2\to C_2-C_1;C_3\to C_3-C_1$
$=\left|\begin{array}{ccc}1& 0& 0\\ 1& x& 0\\ 1& 0& y\end{array}\right|=1(xy-0)-0+0=xy$
Hence, $D$ is divisible by both $x$ and $y$