How to calculate an optimal Bayes estimator for a class sensitive loss function?

Question

The Bayes estimator uses p(x, y) as a probability mass function over (X, Y ) where X = [n] and Y = [k] thus for every x∈X and every y∈Y, p(x, y) = 1. There is a k x k dimension cost matrix, C ∈ [0,∞) and a loss function L_{C} : [k] × [k]→[0, ∞): L_{C}(y, yˆ) := [y != ˆy]Cy,yˆ expressed the class-sensitive loss function, i.e., if we incorrectly predict yˆ and the correct outcome was y we suffer Cy,yˆ loss. How to Derive the Bayes estimator? I have attempted the derivation and obtained 0 as the Bayes' estimator, would you like to check through the working steps?

The image shows my working out to derive the Bayes estimator uses the problem description. I'm expecting a function or a value as the Bayes' estimator.