In an attempt to see if the Matrix is solvable (http://www.drunkmenworkhere.org/129) and to learn recursion further - Saw the opportunity and shot for it.

Matrix.java

import java.io.BufferedReader;
import java.io.FileReader;
import java.io.IOException;


public class Matrix {

    static int[][] matrix = new int[5][5];
    static int max = 0;
    
    public static void main(String[] args) throws IOException {
        BufferedReader bfr = new BufferedReader(new FileReader("m.txt"));
        for (int i = 0; i < 5; i++) {
            String[] s = bfr.readLine().split(" ");
            for (int j = 0; j < 5; j++) {
                matrix[i][j] = Integer.parseInt(s[j]);
            }
        }
        
        for (int x = 0; x < 5; x++) {
            for (int y = 0; y < 5; y++) {
                recursion(x, y);
            }
        }
        System.out.println(max);
    }

    private static void recursion(int x, int y) {
        recursion(x, y, matrix[x][y], startTable(x, y), new int[]{ matrix[x][y], 0 , 0, 0, 0 });
    }
    
    private static void recursion(int x, int y, int value, boolean[][] table, int[] steps) {
        if (value > max) max = value;
        if (value >= 100) {
            System.out.println();
            for (int i = 0; i < 5; i++) {
                System.out.print(steps[i] + " ");
            }
        }
        for (int _x = 0; _x < 5; _x++) {
            for (int _y = 0; _y < 5; _y++) {
                if (table[_x][_y]) continue;
                else {
                    for (int i = 0; i < 5; i++) {
                        if (steps[i] == 0) {
                            steps[i] = matrix[_x][_y];
                            break;
                        }
                    }
                    recursion(_x, _y, value + matrix[_x][_y], fillTable(table, _x, _y), steps);
                }
            }
        }
    }

    private static boolean[][] startTable(int x, int y) {
        boolean[][] blarg = new boolean[5][5];
        for (int i = 0; i < 5; i++) {
            for (int j = 0; j < 5; j++) {
                blarg[i][j] = false;
            }
        }
        
        for (int i = 0; i < 5; i++) {
            blarg[x][i] = true;
        }
        
        for (int i = 0; i < 5; i++) {
            blarg[i][y] = true;
        }
        return blarg;
    }
    
    private static boolean[][] fillTable(boolean[][] table, int x, int y) {
        for (int i = 0; i < 5; i++) {
            table[x][i] = true;
        }
        
        for (int i = 0; i < 5; i++) {
            table[i][y] = true;
        }
        return table;
    }
    
}