Tucker's theorem for almost skew-symmetric matrices and a proof of Farkas' lemma

A real square matrix A is said to be almost skew-symmetric if its symmetric part has rank one. In this article certain fundamental questions on almost skew-symmetric matrices are considered. Among other things, necessary and sufficient conditions on the entries of a matrix in order for it to be almost skew-symmetric are presented. Sums and subdirect sums are studied. Certain new results for the Moore–Penrose inverse of an almost skew-symmetric matrix are proved. An interesting analogue of Tucker's theorem for skew-symmetric matrices is derived for almost skew-symmetric matrices. Surprisingly, this analogue leads to a proof of Farkas' lemma.