class Mat2
{
   // Five ways to store the 2x2 matrix data.

   double a00, a01, a10, a11;    // very fast

   double[] a = new double[4];   // good for large matrices

   double[] r1 = new double[2];  // row vectors
   double[] r2 = new double[2];

   double[] c1 = new double[2];  // column vectors
   double[] c2 = new double[2];  // mathematically accurate

   double[][] a = new double[2][2];  // Not good for matrices!


   // Two ways to implement matrix multiplication.

   Matrix mult(Matrix m)  // does mutating
   {
       // do the calculation mutating this object
       return this; // for chaining
   }

   Matrix tims(Matrix m) // does an allocation
   {
      return new Matrix( has the calculated data in it );
   }
}


   m1.mult(m2).mult(m3); // chained methods don't compile if mult() is a void method

   // m4 = m1 * m2
   // m5 = m1 * m3
   m4 = m1.mult(m2);
   m5 = m1.mult(m3);  // WRONG
   // We need to do the following (if mult() allows chaining).
   m4 = Identity().mult(m1).mult(m2); // need to allocate an identity matrix
   m5 = Identity().mult(m1).mult(m3);
   // If mult() is a void method, then we need to do the following steps.
   m4 = Identity();
   m4.mult(m1);
   m4.mult(m2);
   m5 = Identity();
   m5.mult(m1);
   m5.mult(m3);


   Memory allocation for each kind of matrix storage.

      Matrix
   +----------+
   | a00  a01 |
   | a10  a11 |
   +----------+

      Matrix
   +----------+          double[4]
   |  +----+  |      +---+---+---+---+
   | a| ---|--|----->|   |   |   |   |
   |  +----+  |      +---+---+---+---+
   +----------+


      Matrix
   +----------+      double[2]
   |  +----+  |      +---+---+
   |r1| ---|--|----->|   |   |
   |  +----+  |      +---+---+
   |          |
   |  +----+  |      +---+---+
   |r2| ---|--|----->|   |   |
   |  +----+  |      +---+---+
   +----------+


      Matrix
   +----------+      double[2]
   |  +----+  |      +---+---+
   |c1| ---|--|----->|   |   |
   |  +----+  |      +---+---+
   |          |
   |  +----+  |      +---+---+
   |c2| ---|--|----->|   |   |
   |  +----+  |      +---+---+
   +----------+


      Matrix
   +----------+     double[2][]       double[2]
   |  +----+  |      +-----+          +---+---+
   | a| ---|--|----->| ----|--------->|   |   |
   |  +----+  |      +-----+          +---+---+
   +----------+      | ----|----+
                     +-----+    |     double[2]
                                |     +---+---+
                                +---->|   |   |
                                      +---+---+