Consider a zero-sum game between Player A and Player B. The mn payoff matrix for Player A is A where the element aijin row i and column j represents the payoff that Player A receives when Player A chooses strategy Ai and Player B chooses strategy Bj. The indexes i = 1,...,m and j = 1,...,n where m and n are positive integers.
Let the optimal mixed strategy for Player A be pand let the optimal mixed strategy for Player B be q. Assume the value of the game v is greater than zero.

  1. (i) Write down the Linear Programming (LP) problem for each player. (Proofs are
    not required. However, you should give clear definitions of any extra notation used.)

  1. (ii) Reconsider the payoff matrix with m= n. Assume matrix A is non-singular. Using the LP problems of Part (b)(i), show that

p= v(AT)^{1}u and q= vA^{1}u

where u Rn is the vector whose entries are all 1.

I need help with the second part of this question,

for part i, I get

$ minimise \sum_{i=1}^{m} x_{i} = \frac{1}{v} subject to \sum_{i=1}^{m}a_{ij} x_{i} \geq 1 $