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%\author{Sebastian van Strien (Imperial College)}
%\title{M?PA48  Games and Dynamics: Lecture Notes}
%\date{Autumn 2015}

\begin{document}
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$$M=  \left( \begin{array}{cccc} 
p_1q_1  & p_1(1-q_1) & (1-p_1)q_1 & (1-p_1)(1-q_1) \\ 
p_2q_3 & p_2(1-q_3)  & (1-p_2)q_3 &(1-p_2)(1-q_1) \\
p_3q_2 & p_3(1-q_2) & (1-p_3)q_2 &(1-p_3)(1-q_2) \\
p_4q_4 & p_4(1-q_4) & (1-p_4)q_4 & (1-p_4)(1-q_4)
\end{array} \right)$$


$$D=\det \left( \begin{array}{cccc} 
p_1q_1 -1 & p_1-1 & q_1-1 & 1 \\ 
p_2q_3 & p_2-1 & q_3 & 1 \\
p_3q_2 & p_3 & q_3 & 1 \\
p_4q_4 & p_4 & q_4 & 1 
\end{array} \right) 
$$  
If $p=q=(1,1,0,0)$ then  $D=0$. What other vectors have this property? 





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