Consider any consistent, asymptotically normal estimateǎof an arbitrary rectangular or square matrixA. This article derives an explicit test for the rank ofAand a related test of (semi) definiteness ofA. Potential applications include testing for identification of structural models, testing for the number of state variables in state-space models (including tests for the order of autoregressive moving average (ARMA) processes), consumer demand analysis applications, and testing for the number of factors in factor analysis and related procedures. The test is based on the Gaussian elimination Lower-Diagonal-Upper triangular (LDU) decomposition. The test is illustrated with an empirical application to testing the order of ARMA processes.