  This note contains two remarks. The first remark concerns the extension of
the well-known Cayley representation of rotation matrices by skew symmetric
matrices to rotation matrices admitting -1 as an eigenvalue and then to all
orthogonal matrices. We review a method due to Hermann Weyl and another method
involving multiplication by a diagonal matrix whose entries are +1 or -1. The
second remark has to do with ways of flipping the signs of the entries of a
diagonal matrix, C, with nonzero diagonal entries, obtaining a new matrix, E,
so that E + A is invertible, where A is any given matrix (invertible or not).
