Draw samples from binary, mutually dependent joint distributions

I am drawing a mental blank right now:

I have three quadratic matrices A00, A01 and A11 of size n. Those represent binary, joint bivariate distributions, i.e.

A00[i, j] = P(x_i = 0, x_j = 0),

A01[i, j] = P(x_i = 0, x_j = 1) and

A11[i, j] = P(x_i = 1, x_j = 1).

Those joint distributions are mutually dependent, i.e. A01[i, j] =/= A00[i, i]*A11[j, j].

Furthermore, I have all conditional probabilities stored in matrices B00, B01, B10 and B11 (where B00[i, j] = P(x_i = 0 | x_j = 0), etc.). Not sure if those are needed though.

I want to draw a sample from this joint probability distribution, i.e. I want a n-sized vector v with binary entries 0 and 1.

I am very uncertain if I can simply take the diagonal entries A00[i, i] for i = 1, ..., n as probabilities for each vector entry v[i] independently, since they are not independent. Or is this actually the way to go?

Has anyone an idea how I can draw a sample from this distribution? Thanks a lot in advance!