int[][] array = new int[2][3];
array[0][0] = 1;
array[0][1] = 2;
array[1][0] = 3;
array[1][1] = 4;
array[0][2] = 5;
array[1][2] = 6;

public class Application {

    public static void main(String[] args) {
        int[][] array = new int[2][3];
        array[0][0] = 1;
        array[0][1] = 2;
        array[1][0] = 3;
        array[1][1] = 4;
        array[0][2] = 5;
        array[1][2] = 6;
        int[] newArray = new int[array.length];
        for (int i = 0; i < array.length; i++) {
            for (int j = 0; j < array[i].length; j++) {
                newArray[i] += array[i][j];
            }
        }
        Arrays.stream(newArray).forEach(System.out::println);
    }

}

package array;

/**
 *
 * @author vvm
 */
public class ArrayCount {

    public static void main(String[] args) {
        int[][] array = new int[2][3];
        for (int i = 0; i < 2; i++) {

            for (int j = 0; j < 3; j++) {
                array[i][j] = i+1;

            }
        }
        for (int i = 0; i < 2; i++) {
            System.out.println("");
            for (int j = 0; j < 3; j++) {
                System.out.print(" array[" + i + "]" + "[" + j + "]" + " = " + array[i][j]);
            }
        }
    }
}

int rowsCount = array[0].length;
    for(int row = 0; row < rowsCount; row++)
    {
        int sum = 0;
        for(int i = 0; i < array.length; i++)
        {
            int cell = array[i][row];
            sum += cell;
            // debug (remove next line if not needed!)
            System.out.print(cell + (i < array.length - 1 ? " + " : " = "));
        }

        System.out.println(sum);
    }