We give a direct and simple proof of the equality of Parikh images of lan-
guages generated by matrix grammars with appearance checking with the sets of vectors
generated by register machines. As a particular case, we get the equality of the Parikh
images of languages generated by matrix grammars without appearance checking with
the sets of vectors generated by partially blind register machines. Then, we consider pure
matrix grammars (i.e., grammars which do not distinguish terminal and nonterminal
symbols), and prove the inclusion of the family of Parikh images of languages generated
by such grammars (without appearance checking) in the family of sets of vectors generated by blind register machines, as well as the inclusion of reachability sets of vector
addition systems in the family of Parikh images of pure matrix languages. For pure matrix grammars with a certain restriction on the form of matrices, also the converse of the
latter inclusion is obtained. Thus, in view of the result from, we obtain the semilin-
earity of languages generated by pure matrix grammars (without appearance checking)
with alphabets with at most five letters, with the considered restrictions on the form of
matrices. A pure matrix grammar with five symbols, but without restrictions on the form
of matrices, is produced which generates a non-semilinear language
