Question

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

The correct option is B Divisible by both $x$ and $y$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$

$\Delta =(1+x)(1+y)-1-1(1+y-1)+1(1-1-x)$

$\Delta =(1+x)(1+y)-1-y-x$

$\Delta =xy+1+x+y-1-y-x$

$\Delta =xy$

It is divisible by both $x$ and $y$

$B$ is correct.