In this paper, a new recursive strategy is proposed for determining $\\frac{(n-1)!}{2}$ of $n$th order diagrams. The generalization of $n$th diagram for cross multiplication method were proposed by Pavlovic and Bankier but the specific rule of determining $\\frac{(n-1)!}{2}$ of the $n$th order diagrams for square matrix is yet to be discovered. Thus using combinatorial approach, $\\frac{(n-1)!}{2}$ of the $n$th order diagrams will be presented as $\\frac{(n-1)!}{2}$ starter sets. These starter sets will be generated based on exchanging one element. The advantages of this new strategy are the discarding process was eliminated and the sign of starter set is alternated to each others.<\/p>\r\n","references":"[1] H. Anton, Elementary Linear Algebra, 8th ed. New York: John Wiley, 2000, pp. 1-20.\r\n[2] H. Anton, and R. C. Busby, Contemporary Linear Algebra, New York: John Wiley, 2002, pp. 15-25.\r\n[3] J. D. Bankier, The diagrammatic expansion of Determinants. The American Mathematical Monthly, 68(8), 788-790, 1961.\r\n[4] O. Bretscher, Linear algebra with applications, 4th edition, New Jersey: Prentice Hall International, 2009, pp. 3-15.\r\n[5] C. Y. Hsiung, and G. Y. Mao, Linear algebra. London: World Scientific Publishing, 1998, pp. 3-25.\r\n[6] M. Majahan, and V. Vinay, Determinant: Combinatorics, Algorithms and Complexity. Chicago Journal of Theoretical Computer Science, 730-738, 1997.\r\n[7] S. V. Pavlovic, On The Generalisation of The Sarrus's Rule. Mathematika I Fizika, No. 54, pp. 19-23, 1961. Retrieved on December 2008, form: http:\/\/pefmath2.etf.bg.ac.yu\/files\/40\/54.pdf.\r\n[8] W. L. Perry, Elementary Linear Algebra, New York: McGraw Hill Inc. 1988, pp. 9-13.\r\n[9] K. Sharmila, I. Haslinda, and O. Zurni, Integrated Strategy for generating permutation, Inter. J. of Contemporary Mathematical Sciences, vol. 6, no. 24, pp. 1167-1174.\r\n[10] M. D. Scneider, M. Steeg, and H. F. Young, Linear Algebra, New York: Macmillan Publishing Co., 1982, pp. 3-20.\r\n[12] C. Wilde, Linear Algebra, Massachusetts: Addison-Wesley Publishing Co, 1988, pp. 2-10.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 54, 2011"}