Let the images of the point $A\left(\mathrm{2,3}\right)$ about the lines $y=x$ and $y=mx$ are $P$ and $Q$ respectively. If the line $PQ$ passes through the origin, then $m$ is equal to

Reflection of $A\left(\mathrm{2,3}\right)$ about line $y=x$ is $\left(\mathrm{3,2}\right)=P$
Equation of the line $PQ$ is the same as line $OP$
$\Rightarrow$ equation of the line $PQ$ is $y=\frac{2}{3}x$ and the equation of the line $OA$ is $y=\frac{3}{2}x$.
Now, $y=mx \& y=x$ are the angle bisectors of the lines $2x-3y=0$ and $3x-2y=0$
Now, angle bisectors of $2x-3y=0$ and $3x-2y=0$ are $y=x$ and $y=-x$
$\Rightarrow m=-1$