Some remarks on Hadamard matrices

In this note we use combinatorial methods to show that the unique, up to equivalence, 5 × 5 (1, −1)-matrix with determinant 48, the unique, up to equivalence, 6 × 6 (1,−1)-matrix with determinant 160, and the unique, up to equivalence, 7 × 7 (1,−1)-matrix with determinant 576, all cannot be embedded in the Hadamard matrix of order 8. We also review some properties of Sylvester Hadamard matrices, their Smith Normal Forms, and pivot patterns of Hadamard matrices when Gaussian Elimination with complete pivoting is applied on them. The pivot values which appear reconfirm the above non-embedding results.