  One of the fundamental research problems in the theory of generalized
inverses of matrices is to establish reverse order laws for generalized
inverses of matrix products. Under the assumption that $A$, $B$, and $C$ are
three nonsingular matrices of the same size, the products $AB$ and $ABC$ are
nonsingular as well, and the inverses of $AB$ and $ABC$ admit the reverse order
laws $(AB)^{-1} = B^{-1} A^{-1}$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$,
respectively. If some or all of $A$, $B$, and $C$ are singular, two extensions
of 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 generalized
inverses of the matrices. Thus it is a tremendous work to classify and derive
necessary and sufficient conditions for the reverse order law to hold because
there are all 15 types of $\{i,\ldots, j\}$-generalized inverse for a given
matrix according to combinatoric choices of the four Penrose equations. In this
paper, we first establish several decades of mixed reverse order laws for
$\{1\}$- and $\{1,2\}$-generalized inverses of $AB$ and $ABC$. We then give a
classified investigation to a special family of reverse order laws
$(ABC)^{(i,\ldots,j)} = C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the eight
commonly-used types of generalized inverses using definitions, formulas for
ranges 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.
