A Drazin invertible operator $T \in \mathcal{B}(\mathcal{H})$ is called skew D-quasi-normal operator if $T^*$ and $TT^D$ commute or equivalently $TT^D$ is normal. In this paper, firstly we give a list of conditions on an operator $T,$ each of which is equivalent to $T$ being skew D-quasi-normal. Furthermore, we obtain the matrix representation of these operators. We also develop some basic properties of such operators. Secondly we extend the Kaplansky theorem and the Fuglede-Putnam commutativity theorem for normal operators to skew D-quasi-normal matrices.
