Designs, Codes and Cryptography, 23, 267–281, 2001  C 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Values of Minors of (1, -1) Incidence Matrices of SBIBDs and Their Application to the Growth Problem C. KOUKOUVINOS Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece M. MITROULI Department of Mathematics, University of Athens, Panepistemiopolis 15784, Athens, Greece JENNIFER SEBERRY School of Information Technology and Computer Science, University of Wollongong, Wollongong, NSW, 2522, Australia Communicated by: C. J. Colbourn Received September 8, 1999; Revised May 3, 2000; Accepted May 26, 2000 Abstract. We obtain explicit formulae for the values of the v - j minors, j = 0, 1, 2 of (1, -1) incidence matrices of SBIBD(v, k , λ). This allows us to obtain explicit information on the growth problem for families of matrices with moderate growth. An open problem remains to establish whether the (1, -1) CP incidence matrices of SBIBD(v, k , λ), can have growth greater than v for families other than Hadamard families. Keywords: incidence matrices, SBIBD, minors, Gaussian elimination, growth, complete pivoting 1. Introduction We evaluate the v - j , j = 0, 1, 2 minors for (1, -1) incidence matrices of certain SBIBDs. For the purpose of this paper we will define a SBIBD(v, k , λ) to be a v × v matrix, B , with entries 0 or 1, which has exactly k entries +1 and v - k entries 0 in each row and column and for which the inner product of any distinct pairs of rows and columns is λ. The (1, -1) incidence matrix of B is obtained by letting A = 2 B - J , where J is the v × v matrix with entries all +1. We write I for the identity matrix of order v. Then we have BB T = (k - λ) I + λ J (1) and AA T = 4(k - λ) I + (v - 4(k - λ)) J (2) The determinant simplification theorem (see the Appendix) shows that det B = (k - λ) v-1 2  k + (v - 1)λ = k (k - λ) 1 2 (v-1)