© 2019, © 2019 Taylor & Francis Group, LLC.We consider an extended general linear model containing new observations without making any restrictions on correlation of random vectors and any rank assumptions. We give variety of equalities and inequalities in the comparison of covariance matrix of BLUP of new observations with any other unbiased predictors’ covariance matrices by using rank and inertia formulas. We next consider the general linear mixed model under the assumption that the random effects and the random errors can be correlated. Using connection between the general linear mixed model and the extended general linear model, we give some equalities and inequalities for comparing the covariance matrices of BLUP of linear function of fixed and random effects with covariance matrices of any other type of unbiased predictors. We also give results for special cases and for linear function of partial fixed and random effects.