Why having the loss matrix to be 1 - identity function yields a loss of $1-p(C_l|x)$ ? (Bishop exercise 1.24)

Question

In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:

For a loss matrix $L_{kj}=1-I_{kj}$ we have $\sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$

Now I understand where the $1$ is coming from:

$\sum_kL_{kl}P(C_k|x)=\underbrace{\sum_kp(C_k|x)} - \sum_kp(C_k|x)$

The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?

Answer 1

The second term is actually $\sum_k\color{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.