/// <summary>
/// 遞歸計算行列式的值
/// </summary>
/// <param name="matrix">矩陣</param>
/// <returns></returns>
public static double Determinant(double[][] matrix)
{
  //二階及以下行列式直接計算
  if (matrix.Length == 0) return 0;
  else if (matrix.Length == 1) return matrix[0][0];
  else if (matrix.Length == 2)
  {
    return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
  }
  //對第一行使用“加邊法”遞歸計算行列式的值
  double dSum = 0, dSign = 1;
  for (int i = 0; i < matrix.Length; i++)
  {
    double[][] matrixTemp = new double[matrix.Length - 1][];
    for (int count = 0; count < matrix.Length - 1; count++)
    {
      matrixTemp[count] = new double[matrix.Length - 1];
    }
    for (int j = 0; j < matrixTemp.Length; j++)
    {
      for (int k = 0; k < matrixTemp.Length; k++)
      {
        matrixTemp[j][k] = matrix[j + 1][k >= i ? k + 1 : k];
      }
    }
    dSum += (matrix[0][i] * dSign * Determinant(matrixTemp));
    dSign = dSign * -1;
  }
  return dSum;
}

static void Main(string[] args)
{
  //二階行列式 -2
  double[][] matrix1 = new double[][]
  {
    new double[] { 1, 2 },
    new double[] { 3, 4 }
  };
  Console.WriteLine(Determinant(matrix1));
  //三階行列式 -4
  double[][] matrix2 = new double[][]
  {
    new double[] { 2, 0, 1 },
    new double[] { 1, -4, -1 },
    new double[] { -1, 8, 3 }
  };
  Console.WriteLine(Determinant(matrix2));
  //四階行列式 -21
  double[][] matrix3 = new double[][]
  {
    new double[] { 1, 2, 0, 1 },
    new double[] { 1, 3, 5, 0 },
    new double[] { 0, 1, 5, 6 },
    new double[] { 1, 2, 3, 4 }
  };
  Console.WriteLine(Determinant(matrix3));
  Console.ReadLine();
}