Let \(( \Omega, {\mathcal F}, P)\) be a probability space. This paper focuses on discrete time martingales where the underlying sample space is finite. Suppose \(\Omega= \{ \omega_1, \dots, \omega_s \}\) consists of \(s\) elements and \((M_t)_{t \in T}, T=\{1,2, \dots, n\}\) is a martingale, then construct an \(s \times n\) martingale matrix \(M=[ m_{it}],\) where \(m_{it}= M_t(\omega_i).\) The martingale structure ensures that if \(p_i := P(\omega_i) > 0\) for \(i \leq s\) then the martingale matrix is determined by its last column. The paper explores the basic linear algebraic properties and structure of the class of \(P-\)martingale matrices. Motivated by problems in mathematical finance, related polytopes are also investigated. Let \(\mathbb{P}_M\), a martingale measure polytope, be the set of probability vectors for which a given matrix \(M\) is a martingale matrix. The extreme points of these polytopes are determined.