// using an external function, you'll need to create a new
// matrix object to return.  you don't want to return a
// reference to it.
//matrix& operator+ (const matrix& A, const matrix& B)
matrix operator+ (const matrix& A, const matrix& B)
{
  // you need to instantiate a matrix object to hold the result:
  matrix C;

  // since this is an external function, use the accessor methods
  //for(int ii=0 ; ii<A.nrow ; ii++)
  for(int ii=0 ; ii<A.getRows() ; ii++)
  {
    // ditto for columns
    for(int jj=0 ; jj<A.ncol ; jj++)
    {
      // here, use setElement and getElement
      //p[ii][jj] = A.p[ii][jj] + B.p[ii][jj];
      double element = A.getElement(ii, jj) +
                       B.getElement(ii, jj);
      C.setElement(ii, jj, element);
    }
  }

  // then, you need to return the result
  return C;
}
</pre>

As to your PS, I use the pre tag.  I think the code tag changes to a monospace font, but doesn't preserve white space.