Annals of Fuzzy Mathematics and Informatics Volume 8, No. 1, (July 2014), pp. 19–32 ISSN: 2093–9310 (print version) ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr @FMI c  Kyung Moon Sa Co. http://www.kyungmoon.com Intuitionistic fuzzy incline matrix and determinant Sanjib Mondal, Madhumangal Pal Received 20 December 2011; Accepted 27 February 2013 Abstract. In this paper, we introduce intuitionistic fuzzy incline (IFI), intuitionistic fuzzy incline matrix (IFIM) and its determinant. Also the transitive closure, power of convergent, nilpotence of IFIM and adjoint of an IFIM are considered here. Some properties of determinant of IFIM and triangular IFIM are also introduced here. 2010 AMS Classification: 03E72, 15A15, 15B15 Keywords: Intuitionistic fuzzy incline matrix, Transitive closure, Power of con- vergence, Triangular intuitionistic fuzzy incline matrix. Corresponding Author: Sanjib Mondal (sanjibvumoyna@gmail.com) 1. Introduction Inclines are the additively idempotent semirings in which products are less than or equal to factors. Thus inclines are generalized Boolean algebra, fuzzy algebra and distributive lattice. The Boolean matrices, the fuzzy matrices and the lattice matri- ces are the prototypical examples of the incline matrices. Boolean algebra and fuzzy algebra are applied to automata theory, design of switching circuits, logic of binary relations, medical diagnosis, etc. Marcov chain, information system and clustering are instances in which inclines can be applied. Also, inclines are applied to ner- vous system, probable reasoning, finite state machines, psychological measurement, dynamical programming, decision theory, etc. In 1965, Zadeh [21] developed fuzzy set first, then in 1984 Cao et al. [4] developed incline algebra and its applications. After that several researchers [8, 9, 13, 14, 17] work on this topics. In 1986, Atanassov [1] introduced intuitionistic fuzzy sets (IFS) which becomes a popular topics for investigation in the fuzzy sets community. With max-min operation the fuzzy algebra and its matrix theory are considered by many authors [3, 6, 15, 18, 19]. Determinant theory, powers and nilpotent conditions of matrices over a distributive lattice are considered by Zhang [22] and Tan [20] and the transitivity of matrices over path algebra (i.e., additively idempotent semiring) is discussed by Hashimoto [10, 11, 12]. Generalized fuzzy matrices, matrices over an