feat: Add main method with demonstration example

Author: Raghu0703

src/main/java/com/thealgorithms/matrix/LUDecomposition.java

@@ -74,6 +74,79 @@ public static Result decompose(double[][] matrix) {
         return new Result(l, u);
     }
 
+    /**
+     * Main method for demonstration.
+     *
+     * @param args command line arguments (not used)
+     */
+    public static void main(String[] args) {
+        // Example from the issue
+        double[][] matrix = {
+            {2, -1, -2},
+            {-4, 6, 3},
+            {-4, -2, 8}
+        };
+
+        System.out.println("LU Decomposition Example");
+        System.out.println("========================\n");
+
+        Result result = decompose(matrix);
+
+        System.out.println("Original Matrix A:");
+        printMatrix(matrix);
+
+        System.out.println("\nLower Triangular Matrix L:");
+        printMatrix(result.getL());
+
+        System.out.println("\nUpper Triangular Matrix U:");
+        printMatrix(result.getU());
+
+        // Verify that L * U = A
+        System.out.println("\nVerification: L * U = A");
+        double[][] product = multiplyMatrices(result.getL(), result.getU());
+        printMatrix(product);
+    }
+
+    /**
+     * Helper method to print a matrix.
+     *
+     * @param matrix the matrix to print
+     */
+    private static void printMatrix(double[][] matrix) {
+        for (double[] row : matrix) {
+            System.out.print("{ ");
+            for (int i = 0; i < row.length; i++) {
+                System.out.printf("%.3f", row[i]);
+                if (i < row.length - 1) {
+                    System.out.print(", ");
+                }
+            }
+            System.out.println(" }");
+        }
+    }
+
+    /**
+     * Helper method to multiply two matrices.
+     *
+     * @param a first matrix
+     * @param b second matrix
+     * @return the product matrix
+     */
+    private static double[][] multiplyMatrices(double[][] a, double[][] b) {
+        int n = a.length;
+        double[][] result = new double[n][n];
+
+        for (int i = 0; i < n; i++) {
+            for (int j = 0; j < n; j++) {
+                for (int k = 0; k < n; k++) {
+                    result[i][j] += a[i][k] * b[k][j];
+                }
+            }
+        }
+
+        return result;
+    }
+
     /**
      * Result class to hold L and U matrices from decomposition.
      */