Classification of Linear 2nd-Order PDEs

Hey,

I'm reading a textbook that tells me that if we have a general linear homogeneous pde of 2nd order:

we can create a matrix A, where

and use it to classify the PDE as follows:

Elliptic: Det i.e.

Hyperbolic: Det

Parabolic: Det

which seems fairly simple enough. But in the examples they have, I get something completely different to the solutions they provide.

(a) . I get so Det

(b)

I get and Det

(c)

I get and det

According to the text these should be

(a) Det A = -9/4

(b) Det A = 0

(c) Det A = 5

but I get something different.

Also, as another example, we have

and we should have Det . Where is the coming from?

Thanks