Classification of a system of two second order PDEs with two dependent and two independent variables If we have a second order quasilinear PDE of the form 
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$
where $A,B,C$ are functions of $x,y,u$,
then the equation is called elliptic if $det=\begin{vmatrix}A &C \\C & B\end{vmatrix}>0$, parabolic if $det=0$ and hyperbolic if $det<0$. 
Now what happens if we have a system of two coupled PDEs of the form 
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+D\frac{\partial^2 v}{\partial x^2}+E\frac{\partial^2 v}{\partial y^2}+2F\frac{\partial^2 v}{\partial x\partial y}+ lower\,\, order \,\, terms=0 \\
G\frac{\partial^2 u}{\partial x^2}+H\frac{\partial^2 u}{\partial y^2}+2K\frac{\partial^2 u}{\partial x\partial y}+L\frac{\partial^2 v}{\partial x^2}+M\frac{\partial^2 v}{\partial y^2}+2N\frac{\partial^2 v}{\partial x\partial y}+ lower\,\, order \,\, terms=0$
with $A,B,C,...$ being functions of $x,y,u,v$.
Does it make sense to construct the determinant 
$det=\begin{vmatrix}A &C & D&F\\C & B&F&E\\G &K & L&N\\K &H & N&M\end{vmatrix}$ 
and investigate its sign, or something like that?
 A: The classification of linear PDEs with constant coefficients is based on the principle that any linear transformation of coordinates should preserve the classification. That is, we define
$$
\xi = ax+by+c, 
\quad
\eta = dx+ey+f. 
$$
where $ae \neq bd$ so that lines of constant $\xi$ and $\eta$ are not parallel. Substituting in this change of coordinates, you will find a new system of PDEs in $\xi, \eta$ which have a new set of coefficients. A classification of the system should be the same whether the coefficients of the $x, y$ or $\xi, \eta$ system are used. However, the classes of your system will be more numerous than those of the classical second order PDE.
With the general approach out of the way, I suspect that your system will be classified by the sign of the following 4 determinants:
$$
\left|\begin{array} {ccc} 
A & C \\
C & B
\end{array} \right|
$$
$$
\left|\begin{array} {ccc} 
D & F \\
F & E
\end{array} \right|
$$
$$
\left|\begin{array} {ccc} 
G & K \\
K & H
\end{array} \right|
$$
$$
\left|\begin{array} {ccc} 
L & N \\
N & M
\end{array} \right|
$$
and should be classified by four words, based on the sign of each. You should verify that the signs of these determinants are invariant under the change of coordinates.
Edit:
For the case where the coefficients are not constant, we use exactly the same classification system pointwise. Defining the change of coordinates 
$$
\xi = \xi(x, y), 
\quad
\eta = \eta(x, y) 
$$
which has a nonzero and finite Jacobian at all points, then local to any point $(x_0,y_0)$ we have that
$$
\xi(x, y) \sim \xi(x_0, y_0) + (x-x_0) \cdot \frac{\partial \xi} {\partial x}(x_0, y_0) + (y-y_0) \cdot \frac{\partial \xi} {\partial y}(x_0, y_0)
$$
$$
\eta(x, y) \sim \eta(x_0, y_0) + (x-x_0) \cdot \frac{\partial \eta} {\partial x}(x_0, y_0) + (y-y_0) \cdot \frac{\partial \eta} {\partial y}(x_0, y_0)
$$
which gives us the local linear transform around each point. The coefficients at each point are locally given by there value at each point e.g.
$$
A(x, y, u,v) \sim A(x_0, y_0, u(x_0, y_0),v(x_0, y_0)).
$$
Thus, from an understand of the linear constant coefficient case, the quasi-linear variable coefficient case can be investigated in a pointwise manner. 
