Growth in Gaussian Elimination for Weighing Matrices, W (n, n — 1)

We consider the values for large minors of a skew-Hadamard matrix or conference matrix W of order n and find maximum n x n minor equals to (n — 1)n/2, maximum (n — 1) x (n — 1) minor equals to (n–1)n/2-1 maximum (n — 2) x (n — 2) minor equals to 2(n — 1) n/2–2, and maximum (n — 3) x (n — 3) minor equals to 4(n — 1)n/2-3. This leads us to conjecture that the growth factor for Gaussian elimination of completely pivoted skew-Hadamard or conference matrices and indeed any completely pivoted weighing matrix of order n and weight n — 1 is n — 1 and that the first and last few pivots are (1, 2, 2, 3 or 4, ..., n—1 or n-1/2, n-1/2, n–1) for n > 14. We show the unique W(6, 5) has a single pivot pattern and the unique W(8, 7) has at least two pivot structures. We give two pivot patterns for the unique W(12, 11).