Without expanding, prove that

$\left|\begin{array}{lll}a & b & c \\ x & y & z \\ p & q & r\end{array}\right|=\left|\begin{array}{lll}x & y & z \\ p & q & r \\ a & b & c\end{array}\right|=\left|\begin{array}{lll}y & b & q \\ x & a & p \\ z & c & r\end{array}\right|$

$\left|\begin{array}{lll}x & y & z \\ p & q & r \\ a & b & c\end{array}\right| R_{2} \leftrightarrow R_{3}=-\left|\begin{array}{lll}x & y & z \\ a & b & c \\ p & q & r\end{array}\right| R_{1} \leftrightarrow R_{2}=\left|\begin{array}{lll}a & b & c \\ x & y & z \\ p & q & r\end{array}\right|$

$\left|\begin{array}{lll}y & b & q \\ x & a & p \\ z & c & r\end{array}\right|=\left|\begin{array}{lll}y & x & z \\ b & a & c \\ q & p & r\end{array}\right| C_{1} \leftrightarrow C_{2}=-\left|\begin{array}{lll}x & y & z \\ a & b & c \\ p & q & r\end{array}\right| R_{1} \leftrightarrow R_{2}=\left|\begin{array}{lll}a & b & c \\ x & y & z \\ p & q & r\end{array}\right|$