Alternatives to Mersenne Twister We have Mersenne Twister pseudorandom number generator. It has several variants, CryptMT among them, that is of cryptographic grade.

There are alternatives to Mersenne Twister, like Xorshift or Xoroshiro.

Question: Do those algorithms have cryptographic variants, or are they secure per se, or are there other simple pseudorandom number generators that are good for cryptography?

I need a generator that has good random properties as it will be used to generate random matrices (not keys). At the same time it has to be unpredictable enough so the attacker has little control what matrices are generated when messing with the seed.

CLARIFICATION:

The setup is as follows: a random seed is chosen and used to generate pseudorandom square matrices over Boolean ring. Those matrices, when chosen randomly, have certain probability of being singular or not. The PRNG must generate such a sequence so the probability of a matrix being singular is the same as if they were chosen randomly.

The matrices are not secret, they are presented to the user and forward secrecy is not needed. Based on initial state the user might be able to predict all next matrices in a sequence, as long as they admit the right probability distribution.

ATTACK SCENARIO:

An attacker might be able to alter some bits of the random seed. Not all bits, as he might be restricted to altering only a subset of bits of the seed, or flipping several bits at once. Even when the attacked has such ability, the probability of a matrix being singular should be the same as if it was chosen randomly.

HARDER PROBLEM:

We want not only (non)singularity of a matrix to be close to random distribution, but also its rank.
Forget the Mersenne Twister and xorshift; maybe CryptMT is secure but it has received very little scrutiny since it was rejected from the eSTREAM portfolio a decade ago.  Cryptography is rife with good pseudorandom number generators, like ChaCha: if you pick the seed uniformly at random, the attacker has no hope of distinguishing the output from uniform random, and the world has high confidence in this state of affairs because it has been subjected to heavy scrutiny and there are trillions of euros of economic value protected by ChaCha in HTTPS around the world.

If you have measured performance, and you have profiled, and you have confirmed that a lot of time is spent in ChaCha, and you are confident that this will never be security-relevant—in particular, if you can quantify the statistical tests you need to defeat and argue that smarter statistical tests will never be relevant—then maybe you will have a reason to look elsewhere.  But the first thing you can do with ChaCha is tune down the number of rounds from the default ChaCha20 to the cheaper ChaCha12 or even cheaper ChaCha8.



Update, for question's addendum:

If the adversary can control the seed, then even if you used a random oracle for the PRNG (which is the strongest possible model—not attained by ChaCha but a reasonable model for, say, SHAKE128), the adversary can spend an expected $t$ trials finding a seed with a property that occurs only $1/t$ of the time, like $t = 2^n$ trials to find a seed under which the first $n$ bits of output are all zero.  No amount of cryptography can help you against this.

But if the adversary can't control the seed, the property you want—nonsingular matrices with high probability—is essentially guaranteed with any cryptographic PRNG.  If it weren't, then the putative cryptographic PRNG would be astonishingly bad, because a very simple test—generate a matrix and test whether it's nonsingular—would serve as an attack against the PRNG that breaks it without even knowing how the PRNG is designed!