  In the present paper we propose generalizations of the regularity and
counting lemmas for multidimensional matrices under a finite alphabet. Firstly,
we prove a variant of a multidimensional regularity lemma with the help of a
translation of $\varepsilon$-regularity from graphs to matrices. Next, we state
that this $\varepsilon$-regularity is sufficient for obtaining a matrix
analogue of the counting lemma for $2$-dimensional matrices but not for
higher-dimensional cases. Finally, we introduce $\varepsilon$-regular patterns
that allow us to deduce a multidimensional counting lemma.
