Algorithm Problem (Dynamic Programming): Please provide more detail for learning purposes.

Given a 2-dimensional arrayA[1..m, 1..n], compute the number of ways (paths) of starting at the top-left corner to the bottom-right corner. All moves must go either right or down.

Formulate the recursive relation of the optimal solution; by using the recursive solution

Define the value of an optimal solution recursively in terms of the optimal solutions to subproblems. Let m[i,j] be the minimum number of scalar multiplications needed to compute the matrix Ai s. Thus the cost of cheapest way to compute ALn is m[1,n]. Since computing the product Ai k At+1.; takes P-1PxP; scalar multiplications, where pi is the number of columns of matrix Aj. Thus we have m[i,j] = m[i,k] + m[k+1,j] + P:-1PxP; The overall minimum cost is thus defined by 0 if i=j m[i, j]= min{m[i,k]+m[k+1, j]+ P.-1PxP;} if i