Eigenvalues โ€‹โ€‹- Help classify the first-order PDE system to be elliptic, hyperbolic, or parabolic

The question I need help with is in the picture.

https://i.stack.imgur.com/ynVtD.jpg

I have already added the system to the form
$$
Aq_x + Bq_y = C,
$$

With
$$
A =
begin {bmatrix}
a_1 & 0 \
0 & b_1
end {bmatrix}, quad B =
begin {bmatrix}
0 & a_2 \
b_2 & 0
end {bmatrix}, quad q =
begin {pmatrix}
u \
v
end {pmatrix}.
$$

But I do not know how to proceed from here to find out the eigenvalues โ€‹โ€‹of the term
$$
C =
begin {pmatrix}
g_1 \
g_2
{pmatrix}
$$

that confuses me.
If $ C $ were $ 0 $Then I know how to proceed, but unfortunately that is not the case.

Any help would be appreciated.

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