This problem is part the book I'm using in my game theory course. My work-through is as follows:
Let $A=\begin{pmatrix}3&1&4&0\\1&2&0&5\end{pmatrix}$.
Note how for $P_2$, strategy $(1,0,0,0)$ is strictly dominated by $q=(0,0,\frac{39}{50},\frac{11}{50})$ since $$Aq = (4\cdot\frac{39}{50},5\cdot\frac{11}{50}) = (3\frac{3}{25}, 5\frac1{10})>(3,1)=Ae^1.$$ Similarly, strategy $e^2$ is strictly dominated by $q=(0,0,\frac12,\frac12$), since $Aq=(2,2.5)>(1,2)=Ae^2$.
Here I speed up a bit because it's not as relevant
To solve the remaining game $\begin{pmatrix}4&0\\0&5\end{pmatrix}$, suppose $P_1$'s strategy $p_1=(p,1-p)$. Then we find $4p=5-5p$, so $p=5/9$ with a game value of $20/9$. Solving for $q$ then gives us that $P_2$'s strategy $q=(0,0,5/9,4/9)$.
The problem is, that the book says this is false and that $P_2$'s optimal strategy is $(0,4/5,1/5,0)$, which according to my solution is impossible since $e^2$ is a strictly dominated strategy.
Where exactly am I going wrong here?