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One of the fundamental research problems in the theory of generalizedinverses of matrices is to establish reverse order laws for generalizedinverses of matrix products. Under the assumption that $A$, $B$, and $C$ arethree nonsingular matrices of the same size, the products $AB$ and $ABC$ arenonsingular as well, and the inverses of $AB$ and $ABC$ admit the reverse orderlaws $(AB)^{-1} = B^{-1} A^{-1}$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$,respectively. If some or all of $A$, $B$, and $C$ are singular, two extensionsof the above reverse order laws to generalized inverses can be written as$(AB)^{(i,\ldots,j)} = B^{(i_2,\ldots,j_2)} A^{(i_1,\ldots,j_1)}$ and$(ABC)^{(i,\ldots,j)} = C^{(i_3,\ldots,j_3)}B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$, or other mixed reverse order laws.These equalities do not necessarily hold for different choices of generalizedinverses of the matrices. Thus it is a tremendous work to classify and derivenecessary and sufficient conditions for the reverse order law to hold becausethere are all 15 types of $\{i,\ldots, j\}$-generalized inverse for a givenmatrix according to combinatoric choices of the four Penrose equations. In thispaper, we first establish several decades of mixed reverse order laws for$\{1\}$- and $\{1,2\}$-generalized inverses of $AB$ and $ABC$. We then give aclassified investigation to a special family of reverse order laws$(ABC)^{(i,\ldots,j)} = C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the eightcommonly-used types of generalized inverses using definitions, formulas forranges and ranks of matrices, as well as conventional operations of matrices.Furthermore, the special cases $(ABA^{-1})^{(i,\ldots,j)} =AB^{(k,\ldots,l)}A^{-1}$ are addressed and some applications are presented.

Reverse order laws for generalized inverses of products of two or three matrices with applications