Question

For all real numbers $x,y$ and $z$, the determinant
$\left|\begin{array}{ccc}2x& xy-xz& y\\ 2x+z+1& -xy-xz+yz-z^2& 1+y\\ 3x+1& 2xy-2xz& 1+y\end{array}\right|$ is equal to:

• A. $zero$
• B. $(x-y)(y-z)(z-x)$
• C. $(x-yz)(y-z)$
• D. $(y-xz)(z-x)$

Answer 1

Answer: (B)

$(x-y)(y-z)(z-x)$

$\left|\begin{array}{ccc}2x& xy-xz& y\\ 2x+z+1& -xy-xz+yz-z^2& 1+y\\ 3x+1& 2xy-2xz& 1+y\end{array}\right|$

$=\left|\begin{array}{ccc}2x& x(y-z)& y\\ z+1& z(y-z)& 1\\ x+1& x(y-z)& 1\end{array}\right|$ $\left\{\begin{array}{c}\because R_2\to R_2-R_1\\ R_3\to R_3-R_1\end{array}\right\}=(y-z)$ $\left|\begin{array}{ccc}2x& x& y\\ z+1& z& 1\\ x+1& x& 1\end{array}\right|$

$=(y-z)\left|\begin{array}{ccc}x& x& y\\ 1& z& 1\\ 1& x& 1\end{array}\right|\{\because c_1\to c_1-c_2\}=(y-z)\left|\begin{array}{ccc}x& x& y\\ 0& z-x& 0\\ 1& x& 1\end{array}\right|\{R_2\to R_2-R_3\}$

$\Rightarrow (x-y)(y-z)(z-x)$