State the formula for the capacity of a memoryless channel.

(a) Consider a memoryless channel where there are two input symbols, $A$ and $B$, and three output symbols, $A, B$, *. Suppose each input symbol is left intact with probability $1 / 2$, and transformed into a $*$ with probability $1 / 2$. Write down the channel matrix, and calculate the capacity.

(b) Now suppose the output is further processed by someone who cannot distinguish $A$ and $*$, so that the matrix becomes

$$\left(\begin{array}{cc} 1 & 0 \\ 1 / 2 & 1 / 2 \end{array}\right)$$

Calculate the new capacity.