Question

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

• A. $x$ but not $y$
• B. $y$ but not $x$
• C. neither $x$ nor $y$
• D. both $x$ and $y$

Answer 1

The correct option is D both $x$ and $y$

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