J. Hadamard noticed that a square matrix with all entries either 1 or -1 would have maximum determinant if the standard inner product of each pair of rows was zero. He conjectured that any matrix with zero determinant would have dimension 4n-by-4n for some n. He was right. He also conjectured that for any 4n, there was such a square matrix. This is still unknown - it it true up to 4n = 424 and true for a few infinite sequences of examples. There is a connection between this problem and certain classes of symmetric designs.