 Solution

Every row of the channel matrix is a permutation of $(1-2\alpha ,\alpha ,\alpha )$, and every column sum is one. It is therefore a weakly symmetric channel. The uniform input makes the output uniform, with entropy ${\log }_{2}3$, while the conditional output entropy is the entropy of one row:The weakly symmetric channel capacity formula now gives the ternary symmetric channel capacitybits per channel use, for $0\le \alpha \le 1/2$, with the convention $0{\log }_{2}0=0$.