Urgent help needed: Condition for unique solution for matrix A that have infinite solutions under maximum trace(A)

[zeeshas901]

Hello! 

I need urgent help regarding the following question.

Any help will be greatly appreciated.

Thank you!

Edited February 16, 2020 by zeeshas901
question is added as an attachment

[taeto]

I do not see an immediate solution. But is it not just a linear programming (LP) problem? Clearly you want to maximize a linear function of the matrix entries subject to several linear equality and inequality conditions, the nonnegativity, sum of row entries, and the condition \(b_1A=b_2\) which is equivalent to \(n\) linear equations. So unless I am missing something, the solution will just be standard. 

Edit: I notice that linear programming is already one of the tags for the question posed on a different site. Have you tried to look at the dual linear program (DLP) to the given LP?

Edited February 16, 2020 by taeto

[taeto]

I still do not know if this is a homework question. But let us try a small example.

If we have \(b_1 = (1/2,1/2)\) and \(b_2 = (2/3,1/3),\) do you know how to determine the matrix \(A?\) Do you see why trace\((A) = 5/3 \) is the optimum and \(A\) is uniquely determined?

Edited February 16, 2020 by taeto