ant $6$.  If I swap the order of the two vectors in the
  matrix, then the determinant of
  $\begin{bmatrix}1&3\\2&0\end{bmatrix}$ is $-6$.  The reason for the
  difference is that the determinant not only keeps track of area, but
  also order. Starting at the first vector, if you can turn
  counterclockwise through an angle smaller than 180$^\circ$ to obtain
  the second vector, then the determinant is positive.  If you have to
  turn clockwise instead, then the determinant is negative.  This is
  often te