It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $A_1{B}_1^{g_1}{C}_1+A_2{B}_2^{g_2}{C}_2+\cdots +A_k{B}_k^{g_k}{C}_k=D$ and $A_1{B}_1^{g_1}{A}_2{B}_2^{g_2}\cdots A_k{B}_k^{g_k}{A}_{k+1}=A$, where $A_1$, $A_2$, $\dots$, $A_{k+1}$, $C_1$, $C_2$, $\dots$, $C_k$ and $A$ and $D$ are given, and $B_1^{g_1}$, $B_2^{g_2}$, $\dots$, $B_k^{g_k}$ are generalized inverses of matrices $B_1$, $B_2$, $\dots$, $B_k$. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $\{1\}$- and $\{1,2\}$-generalized inverses of matrices with $k=2,3$ by using some elementary methods, including a series of explicit rank equalities for block matrices.

This article explores the mathematical and statistical performances and connections of the two well-known ordinary least-squares estimators (OLSEs) and best linear unbiased estimators (BLUEs) of unknown parameter matrices in the context of a multivariate general linear model (MGLM) for regression, both of which are defined under two different optimality criteria. Tian and Zhang ^[38] once collected a series of existing and novel identifying conditions for OLSEs to be BLUEs under general linear models: On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model, Stat. Probabil. Lett., 112 (2016), 105–112. In this paper, we show how to extend this kind of results to multivariate general linear models. We shall give a direct algebraic procedure to derive explicit formulas for calculating the OLSEs and BLUEs of parameter spaces in a given MGLM, discuss the relationships between OLSEs and BLUEs of parameter matrices in the MGLM, establish many algebraic equalities related to the equivalence of OLSEs and BLUEs, and give various intrinsic statistical interpretations about the equivalence of OLSEs and BLUEs of parameter matrices in a given MGLM using some matrix analysis tools concerning ranks, ranges, and generalized inverses of matrices.