  We give a general definition of Manin matrices for arbitrary quadratic
algebras in terms of idempotents. We establish their main properties and give
their interpretation in terms of the category theory. The notion of minors is
generalised for a general Manin matrix. We give some examples of Manin
matrices, their relations with Lax operators and obtain the formulae for some
minors. In particular, we consider Manin matrices of the types $B$, $C$ and $D$
introduced by A. Molev and their relation with Brauer algebras.
Infinite-dimensional Manin matrices and their connection with Lax operators are
also considered.
