 Solution

The channel matrix has entries $M_{xy}=P(Y=y\mid X=x)$, with nonnegative entries and each row summing to one. For an input distribution $\pi$, it induces the joint law $P(X=x,Y=y)={\pi }_x{M}_{xy}$. Its capacity in bits per use isOn finite alphabets the maximum exists by continuity on the compact probability simplex.