Let A be a real m×n matrix, with no two entries equal.
Let A(i,j) be the matrix entry obtained by selecting the least entry in each row, and then the greatest of these entries. Let A(u,v) be obtained by selecting the largest entry from each column, and then the least entry from these.
Prove that A(i,j) ≤ A(u,v)
Example:
A = \begin{pmatrix}
3 & 52 & 32 \\
34 & 66 & 26
\end{pmatrix}.
A(i,j) = 26
A(u,v) = 32\
EDIT
My effort: I flattened the Matrix and sorted the entries. For both, A(i,j) and A(u,v), there must be at least (m-1) entries less than them, and at least (n-1) entries greater than them. This means that our value lies somewhere between mth and (mn-n+1)th entry (of the flattened/sorted matrix). However, I still can't prove that A(i,j) ≤ A(u,v)