In this paper, we delineate the results to present a comprehensive methodology for an over-parameterized linear model. This model is derived by augmenting new regressors to a linear model. The analysis includes an over-parameterized linear model, its appropriately reduced model, and a base linear model of the over-parameterized model. The predictors and estimators known as the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) are discussed in relation to their analytical expressions and qualities within the context of these three models. We analyze the comparison problems regarding the dispersion matrices of the BLUPs/BLUEs inside specified models utilizing matrix algebra. Specifically, we utilize formulas for matrix rank and inertia, along with simple block matrix operations. Specifically, we establish many equations and inequalities to compare the dispersion matrices of the BLUPs/BLUEs of a general vector that contains joint unknown vectors under the given models.