Question

Let ${\Delta }_1=\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$,
${\Delta }_2=\left|\begin{array}{ccc}1& x& y\\ 1& x+y& y\\ 1& x& x+y\end{array}\right|$,
${\Delta }_3=\left|\begin{array}{ccc}x& x+y& x+2y\\ -x& x& 0\\ 0& -x& x\end{array}\right|$,
${\Delta }_4=\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& 1& 1\end{array}\right|$
then

Answer 1

A) ${\Delta }_1=\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$
Applying $C_1\to C_1+C_2+C_3$, we get
${\Delta }_1=\left|\begin{array}{ccc}2(x+y)& y& x+y\\ 2(x+y)& x+y& x\\ 2(x+y)& x& y\end{array}\right|$
Again applying $R_2\to R_2-R_1,R_3\to R_3-R_1$, we get
${\Delta }_1=\left|\begin{array}{ccc}2(x+y)& y& x+y\\ 0& x& -y\\ 0& x-y& -x\end{array}\right|=2(x+y)(-x^2+y(x-y))=-2(x^3+y^3)$
B) ${\Delta }_2=\left|\begin{array}{ccc}1& x& y\\ 1& x+y& y\\ 1& x& x+y\end{array}\right|$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$, we get
${\Delta }_2=\left|\begin{array}{ccc}1& x& y\\ 0& y& 0\\ 0& 0& x\end{array}\right|=1(xy-0)=xy$
C) ${\Delta }_3=\left|\begin{array}{ccc}x& x+y& x+2y\\ -x& x& 0\\ 0& -x& x\end{array}\right|$
Applying $R_2\to R_2+R_1$, we get
${\Delta }_3=\left|\begin{array}{ccc}x& x+y& x+2y\\ 0& 2x+y& x+2y\\ 0& -x& x\end{array}\right|=x((2x+y)x+x(x+2y))=x(2x^2+xy+x^2+2xy)=3x^2(x+y)$
D) ${\Delta }_4=\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& 1& 1\end{array}\right|=1(y-y^2)-1(x-x^2)+1(x{y}^2-y{x}^2)=(x-y)(y-1)(1-x)$