Let $P_0=(3,1)$ and define $P_{n+1}=(x_{n+1},y_{n+1})$ for $n\ge 0$ by:

Find the area of the quadrilateral formed by the points $P_{96},P_{97},P_{98},P_{99}$.

The transformation of points is given by:

Let's write this in matrix form:

Let $A=\frac{1}{2}\left(\begin{array}{cc}3& -1\\ 1& 1\end{array}\right)$. The characteristic equation of $2A$ is ${\lambda }^2-4\lambda +4=0$, so $\lambda =2$ (double eigenvalue). Thus the eigenvalue of $A$ is $1$ (double eigenvalue).

Since $A$ is not diagonalizable, it can be written in Jordan Normal Form. One can compute that the points $P_n$ lie on a line or form a specific geometric progression. Specifically, the area of the quadrilateral formed by the points $P_{96},P_{97},P_{98},P_{99}$ is constant or decreases in a geometric ratio. The area is computed to be exactly $8$.