# linear algebra – Property of the number of these integer matrices.

Given a positive integer $$N$$, to put
$$mathcal {A} = { begin {pmatrix} a & b \ c & d end {pmatrix} : a, b, c, d in mathbb {Z} ,, , ad-bc = 4 ,, , | a |, | b |, | c |, | d | N }$$
and
$$mathcal {B} = { begin {pmatrix} a & b \ c & d end {pmatrix} : a, b, c, d in mathbb {Z} ,, , ad-bc = 4 ,, , | a |, | b |, | c |, | d | n ,, , a, b, c, d text {are even} }$$

Denote $$a = # mathcal {A}$$ and $$b = # mathcal {B}$$. The question is how to evaluate $$a$$ and $$b$$ for given $$N$$ (Just suppose that $$N = 10000$$) and $$lim_ {N to infty} frac {b} {a} =?$$

My attempt

I thought that for $$mathcal {B}$$ it is equivalent to calculating the number of these integer matrices whose determinant is $$1$$. And I wanted to use Smith's normal form but I found myself stuck on how to use the condition of the variable range.