Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we investigate Arens regularity and weak amenability of certain matrix algebras which are built on matrix Banach algebras. In particular we show that for such algebras both of Arens regularity and weak amenability problems can be reduced to the same problem for a considerably smaller algebra.