The maximal determinant and subdeterminants of \(\pm\)1 matrices. (Q1414147)

A matrix of order \(n\) is said to be an Hadamard matrix if all its entries are \(\pm 1\) and its distinct row and column vectors are orthogonal. The authors investigate \(\pm 1\)-matrices with maximal absolute values of determinants, especially Hadamard matrices, and their submatrices. It is proved that if \(H\) is a \(12 \times 12\)-Hadamard matrix then a square \(\pm 1\)-matrix of size 5 with a maximal absolute value of the determinant can be embedded to \(H\). Several conjectures concerning the numbers that can appear as maximal values of determinants for \(\pm 1\)-matrices are posed and verified for small values of \(n\). Among them there is a conjecture that the determinant of a \(\pm 1 \)-matrix of order \(n\) is a multiple of \(2^{n-1}\). Principal minors occurring in a completely pivoted \(\pm 1\)-matrix are studied. In particular, it is proved that every Hadamard matrix of order \(n\geq 4\) contains a submatrix which can be obtained from the matrix \(A= E_{11}+E_{12}+E_{13}+E_{14}+E_{21}-E_{22}+E_{23}-E_{24}+ E_{31}+E_{32}-E_{33}-E_{34}+E_{41}-E_{42}-E_{43}+E_{44}\) by permutation of rows/columns and multiplication of rows/columns with \(\pm 1\). It is shown that every completely pivoted Hadamard matrix contains \(A\) in its left upper corner. An algorithm to compute \((n-j)\times (n-j)\) minors of Hadamard matrices of order \(n\) is presented and these minors are determined for \(j=1,2,3,4\).