Please guide me on what conclusions can I draw from the following information
Nine people are standing in a $3\times3$ matrix arrangement. Rajat is the tallest among the shortest persons in each row. Sanket is the shortest among the tallest persons in each column
A. Rajat is the 3rd shortest person
B. Sanket is shorter than Rajat
C. Sanket is the 3rd tallest person.
D. Sanket is taller than Rajat
The only thing I was able to think of was writing
Rajat${}= \max( \min(a_{11},a_{12},a_{13}) , \min(a_{21},a_{22},a_{23}) , \min(a_{31},a_{32},a_{33}) )$
Sanket${} = \min( \max(a_{11},a_{21},a_{31}) , \max(a_{12},a_{22},a_{32}) , \max(a_{13},a_{23},a_{33}) )$
Let's assume all $9$ people are of different lengths, and let's represent their lengths by $1$ through $9$ ($9$ being the tallest)
Now consider the following arrangement:
\begin{array}{ccc} 7 &1&4\\ 5&8&2\\ 3&6&9\\ \end{array}
With this arrangement we find that Rajat is of length $max(min(7,1,4),min(5,8,2),min(3,6,9))=max(1,2,3)=3$ and Sanket is of length $min(max(7,5,3),max(1,8,6),max(4,2,9))=min(7,8,9)=7$, and so claims A,C, and D are true and B is false.
Now consider the following arrangement:
\begin{array}{ccc} 1&7&5\\ 2&8&3\\ 6&4&9\\ \end{array}
For this arrangement we find that Rajat is of length $max(min(1,7,5),min(2,8,3),min(6,4,9))=max(1,2,4)=4$, and Sanket of length $min(max(1,2,6),max(7,8,4),max(5,3,9))=min(6,8,9)=6$. So, this time, claims A and C are false, but B is still false, and D is still true.
The logic of the solution is correct: D is always true, and therefore B is always false. But as the two scenarios above show, A and C can be either true or false.