Question

Which one of the following does NOT equal
$\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|$ ?

• A. $\left|\begin{array}{ccc}1& x(x+1)& (x+1)\\ 1& y(y+1)& (y+1)\\ 1& z(z+1)& (z+1)\end{array}\right|$
• B. $\left|\begin{array}{ccc}1& x+1& x^2+1\\ 1& y+1& y^2+1\\ 1& z+1& z^2+1\end{array}\right|$
• C. $\left|\begin{array}{ccc}0& x-y& x^2-y^2\\ 0& y-z& y^2-z^2\\ 1& z& z^2\end{array}\right|$
• D. $\left|\begin{array}{ccc}2& x+y& x^2+y^2\\ 2& y+z& y^2+z^2\\ 1& z& z^2\end{array}\right|$

Answer 1

The correct option is A $\left|\begin{array}{ccc}1& x(x+1)& (x+1)\\ 1& y(y+1)& (y+1)\\ 1& z(z+1)& (z+1)\end{array}\right|$

Let us verify option (A):
$\left|\begin{array}{ccc}1& x^2+x& x+1\\ 1& y^2+y& y+1\\ 1& z^2+z& z+1\end{array}\right|=\left|\begin{array}{ccc}1& x^2+x& x\\ 1& y^2+y& y\\ 1& z^2+z& z\end{array}\right|+\left|\begin{array}{ccc}1& x^2+x& 1\\ 1& y^2+y& 1\\ 1& z^2+z& 1\end{array}\right|$

$=\left|\begin{array}{ccc}1& x^2& x\\ 1& y^2& y\\ 1& z^2& z\end{array}\right|+\left|\begin{array}{ccc}1& x& x\\ 1& y& y\\ 1& z& z\end{array}\right|+0$
$=-\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|+0\ne \left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|$