  In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for
any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$
denote the usual transposition. In this work, we explore the possibility of
deriving the matrix equality $(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ for any $4
\times 4$ matrices $A$ and $B$, where $\Gamma$ denote the partial
transposition. We found that, in general, $(AB)^{\Gamma}\neq
A^{\Gamma}B^{\Gamma}$ holds for $4 \times 4$ matrices $A$ and $B$ but there
exist particular set of $4 \times 4$ matrices for which $(AB)^{\Gamma}=
A^{\Gamma}B^{\Gamma}$ holds. We have exploited this matrix equality to
investigate the separability problem. Since it is possible to decompose the
density matrices $\rho$ into two positive semi-definite matrices $A$ and $B$ so
we are able to derive the separability condition for $\rho$ when
$\rho^{\Gamma}=(AB)^{\Gamma}=A^{\Gamma}B^{\Gamma}$ holds. Due to the
non-uniqueness property of the decomposition of the density matrix into two
positive semi-definte matrices $A$ and $B$, there is a possibility to
generalise the matrix equality for density matrices lives in higher dimension.
These results may help in studying the separability problem for higher
dimensional and multipartite system.
