Derive the criterion for minimizing the expected loss when there is a general loss matrix and general prior probabilities for the classes

Question

In the book "Pattern Recognition and Machine Learning" I am trying to do exercise 1.23 (p.63):

Derive the criterion for minimizing the expected loss when there is a general loss matrix and general prior probabilities for the classes.

On p.42 it states that (given a loss matrix $L$ which entries $L_{kj}$ denote the loss of assigning an observation $x$ to class $j$ when in reality it's in class $k$) we need to assign an observation $x$ to the class that minimizes the quantity $$\sum_kL_{kj}p(C_k|x)$$

So to answer the question:

We have that $p(C_k|x)=\frac{p(x|C_k)p(C_k)}{p(x)}$, we can express the quanity we want to minimize in terms of $p(C_k)$:

$$\sum_kL_{kj}p(C_k|x)=\sum_kL_{kj}\frac{p(x|C_k)p(C_k)}{p(x)}\propto \sum_kL_{kj}p(x|C_k)p(C_k)$$

So we assign an observation $x$ to $j$ for which $\sum_kL_{kj}p(x|C_k)p(C_k)$ is minimized.

Is this what the question is asking for?