  For two positive definite adjointable operators $M$ and $N$, and an
adjointable operator $A$ acting on a Hilbert $C^*$-module, some properties of
the weighted Moore-Penrose inverse $A^\dag_{MN}$ are established. When
$A=(A_{ij})$ is $1\times 2$ or $2\times 2$ partitioned, general representations
for $A^\dag_{MN}$ in terms of the individual blocks $A_{ij}$ are studied. In
case $A$ is $1\times 2$ partitioned, a unified representation for $A^\dag_{MN}$
is presented. In the $2\times 2$ partitioned case, an approach to constructing
Moore-Penrose inverses from the non-weighted case to the weighted case is
provided. Some results known for matrices are generalized in the general
setting of Hilbert $C^*$-module operators.
