On the maksimum determinants of matrix that some (0,1) and (-1,1)

In this thesis, firstly; information about the definition and representation of matrix is given. Various matrix types including mainly Hessenberg matrix and Hadamard matrix are introduced. Basic definitions and theories on Determinant function and Fibonacci numbers are emphasized. The general definition of Determinant function is given by using then features of second and third degree determinants. A different method of determinant calculating called "The Dodgson Condensation method" which was introduced in an essay in 1866 by Charles Lutwidge Dodgson, the author of Alice in Wonderland, is explained in the thesis. It is explained how to use the Dodgson condensation method in solving linear equation systems. Then, it is indicated that the determinant of a lower Hessenberg matrix which has special conditions is equal with the Fibonacci number corresponding to the degree of matrix. The relation between the maximum determinant of any n×n lower Hessenberg matrices and Fibonacci numbers is examined. The maximum determinants of matrices which have ±1 elements and special conditions are investigated. In the last chapter, BIBD, SBIBD designs and incidence matrices are introduced. For all (-1, 1) matrices, the maximum determinant boundaries or the lower bounds values for maximum determinant are examined using equivalent SBIBD results.