See http://stackoverflow.com/questions/32875905/maximum-cost-of-traversal-in-matrix-using-dynamic-programming?noredirect=1#comment53701431_32875905

MatrixTraversal.java

package stackoverflow;

import java.util.Scanner;

public class MatrixTraversal {

    static int[][] cost;
    static int m, n, maxCost = 0;

    public static void solve(int currRow, int currCol, int[][] isVisited, int currCost) {

        int upperRow, lowerRow, rightCol;

        isVisited[currRow][currCol] = 1;            //marking current position in in matrix as visited

        currCost += cost[currRow][currCol];             //total cost upto current position

        if( currCol == (n - 1)                          //if we have reached the last column in matrix
            && maxCost < currCost )                     //and present cost is greater than previous maximum cost
            maxCost = currCost;

        upperRow = ((currRow - 1) + m) % m;             //upper row value taking care of teleportation
        lowerRow = (currRow + 1) % m;                   //lower row value taking care of teleportation
        rightCol = currCol + 1;                         //right column value

        if( isVisited[upperRow][currCol] == 0 )     //if upper cell has not been visited
            solve(upperRow, currCol, isVisited, currCost);

        if( isVisited[lowerRow][currCol] == 0 )     //if lower cell has not been visited
            solve(lowerRow, currCol, isVisited, currCost);

        if( rightCol != n &&                            //if we are not at the last column of the matrix
            isVisited[currRow][rightCol] == 0 )     //and the right cell has not been visited
            solve(currRow, rightCol, isVisited, currCost);

        isVisited[currRow][currCol] = 0;
    }

    public static void main(String[] args) {

        int[][] isVisited;
        int i, j;

        Scanner sc = new Scanner(System.in);

        System.out.print("Enter the no.of rows(m): ");
        m = sc.nextInt();

        System.out.print("Enter the no.of columns(n): ");
        n = sc.nextInt();

        cost = new int[m][n];
        isVisited = new int[m][n];

        System.out.println("Enter the cost matrix:");
        for(i = 0; i < m; i++)
            for(j = 0; j < n; j++)
                cost[i][j] = sc.nextInt();              //generating the cost matrix

        for(i = 0; i < m; i++)
            solve(i, 0, isVisited, 0);                  //finding maximum traversal cost starting from each cell in 1st column 

        System.out.println(maxCost);

    }

}