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Abstract. Intensive investigation on various theoretical properties of Parikh matrices has been taken place in the theory of formal languages. This paper deals with the problem of finding properties of words so that their Parikh matrices and Anti-diagonal matrices commute.

A word, mathematically expressed, is a sequence of symbols in a finite set, called an alphabet. Parikh matrix is an ingenious tool providing information on certain subsequences of a word, referred to as subwords. On the other hand, based on subwords of a word, the notion of precedence matrix or p-matrix of a word has been introduced in studying a property, known as fair words. In this paper we consider p-matrix for words especially over binary and ternary alphabets and obtain several algebraic properties of the p-matrix.