Use properties of matrix addition and multiplication to deduce from Problem 25 that if P^* and Q^* are optimal strategies for the game M with value v, then they are also optimal strategies for the game M + kJ with value v + k.

Data from Problem 25

Let P and Q be strategies for the 2 × 2 matrix game M. Let k be a constant, and let J be the matrix with all 1’s as entries. Show that the matrix product P(kJ) Q equals the 1 × 1 matrix k.

Generalize part (A) to the situation where M and J are m × n matrices.