C++: Creating Matrix template using vector

I want to create a Matrix template that uses vector. For the time being I want to create the following operations. I need setting the implementation for the operations.
Maybe I do not have to use a pointer either.

template <class T>
class Matrix {

protected:

  typedef vector<T>*  VP;

  int  M;    ///< Number of rows
  int  N;    ///< Number of column
  VP*  MAT;  ///< Matrix object (pointer to vector)

public:

  ////////////////////////////////////////////////
  /// \brief Creates a matrix
  /// Matrix b;

  Matrix
  (
   );

  ////////////////////////////////////////////////
  /// \brief Creates a matrix of size (m,n), 
  /// Matrix b(3,2);

  Matrix
  (
     const int  m,
     const int  n
   );

  ////////////////////////////////////////////////
  /// \brief Sets a matrix to another matrix
  /// Matrix b(a);

  Matrix
  (
     const Matrix&  a
   );

  ////////////////////////////////////////////////
  /// \brief Destroys a matrix

  ~Matrix 
  (
   );

};


template <class T>
inline 
Matrix<T>::Matrix
(
) {

  M = N = 0;

}


template <class T>
inline
Matrix<T>::Matrix
(
 const int  m,
 const int  n
 ) {

  M = m;
  N = n;
  MAT = new VP[M];

  for (int i = 0;  i < M;  i++)
    {
      MAT = new vector<T> (N);
    }

}


template <class T>
inline
Matrix<T>::Matrix
(
 const Matrix<T>&  a
 ) {

  M = a.M;
  N = a.N;
  MAT = new VP[M];

  for (int i = 0;  i < M;  i++)
    {
      MAT = new vector<T> (N);
    }

}

Why not use the already existing vector class?

That's what I want to do. I have some code that uses its own vector class. Now want to change it to use std:: vector

---------- Post updated at 10:46 AM ---------- Previous update was at 10:24 AM ----------

Have now started modifications to use vector.

Current problem is what to do instead of calling

MAT = new vector<T> (N);

template <class T>
class Matrix {

private:
  std::vector< std::vector<T> > MAT;

protected:

  unsigned  M;  ///< Number of rows
  unsigned  N;  ///< Number of column

public:

  ////////////////////////////////////////////////
  /// \brief Creates a matrix
  /// Matrix b;

  Matrix
  (
   );

  ////////////////////////////////////////////////
  /// \brief Creates a matrix of size (m,n), 
  /// Matrix b(3,2);

  Matrix
  (
     const int  m,
     const int  n
   );

  ////////////////////////////////////////////////
  /// \brief Sets a matrix to another matrix
  /// Matrix b(a);

  Matrix
  (
     const Matrix&  a
   );

  ////////////////////////////////////////////////
  /// \brief Destroys a matrix

  ~Matrix 
  (
   );

  ////////////////////////////////////////////////
  /// \brief      Return a reference to element i in a vector
  /// \param[in]  i

  Vector<T>&  
  operator [] 
  (
     const int  i
   );

  ////////////////////////////////////////////////
  /// \brief      Return a reference to element i in a vector
  /// \param[in]  i

  vector<T>&  
  operator [] 
  (
     const int  i
   ) const;


};

---------- Post updated at 11:01 AM ---------- Previous update was at 10:46 AM ----------

Not sure how much this will work

template <class T>
inline

Matrix<T>::Matrix
(
 const unsigned  m,
 const unsigned  n
 ) {

  M = m;
  N = n;

  for (unsigned i = 0;  i < M;  i++)
    {
      MAT.resize (N);
    }

}


template <class T>
inline

Matrix<T>::Matrix
(
 const Matrix<T>&  a
 ) {

  M = a.M;
  N = a.N;

  for(int i = 0; i<M; i++)
    {
      vector<T>  v;
      for(int j = 0; j<N; j++)
        {
          v.push_back(a[j]);
        }
      MAT.push_back(v);
    }

}