probleme to save object into Xml file c#

I'm trying to save a dictionary of Matrix into a Xml file. My Matrix class attributes are :

 public class Matrix 
{
    public int Lines          { get; set; }
    public int Columns        { get; set; }
    public double[,] Elements { get; set; }
    public string name        { get; set; }
 }

After many attempts, i wrote this :

                string fileName = dlg.FileName;
                Stream writer = new FileStream(fileName,FileMode.Create);

                foreach (KeyValuePair<String, Matrix> matrice in CalMat.Calculatrice.listMatrix)
                {
                    XmlSerializer x = new XmlSerializer(matrice.GetType());
                    x.Serialize(writer, matrice);

                }
                writer.Close();

If i run this code with one matrix, the file is created, but i only have this sentence written :

<?xml version="1.0"?><KeyValuePairOfStringMatrix xmlns:xsi="http://ift.tt/ra1lAU" xmlns:xsd="http://ift.tt/tphNwY" /><?xml version="1.0"?> 

I think my code is missing something but I don't know what. A write method, I guess.

Thank you for your time!