= 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:
$$
H(Y\mid X)
=-(1-2\alpha)\log_2(1-2\alpha)
-2\alpha\log_2\alpha.
$$
The <weakly symmetric channel capacity> formula now gives the <ternary symmetric channel capacity>
$$
\boxed{
C=\log_2 3
+(1-2\alpha)\log_2(1-2\alpha)
+2\alpha\log_2\alpha
}
$$
bits per channel use, for $0\leq\alpha\leq1/2$, with the convention $0\log_2 0=0$.
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