Question

If $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right]$ for $x\ne 0,y\ne 0$, then $D$ is:

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

Answer 1

Answer: (B)

divisible by both $x$ nor $y$

$D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$
$=1\left|\begin{array}{cc}1+x& 1\\ 1& 1+y\end{array}\right|-1\left|\begin{array}{cc}1& 1\\ 1& 1+y\end{array}\right|+1\left|\begin{array}{cc}1& 1+x\\ 1& 1\end{array}\right|$
$=(1+x)(1+y)-1-(1+y)+1+1-(1+x)$
$=(1+x)(1+y)-1-y+1-1-x$
$=(1+x)(1+y)-x-y-1$
$=1+y+x+xy-x-y-1$
$=xy$
$\therefore$ $D$ is by both $x$ and $y$.