Linear Transformations · Superchallenge problem

An invertible linear map $T:\mathbb{R}^2\to\mathbb{R}^2$ maps the line $y = x$ onto the line $y = 2x$, the line $y = 2x$ onto the line $y = 3x$, and the line $y = 3x$ onto the line $y = x$. Find $\dfrac{\det T}{(\operatorname{tr} T)^2}$.