From the general linear model, we know the following :

$E(\pmb{\hat{\beta}}) = E((\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}) $

and

$Var(\pmb{\hat{\beta}}) = \sigma^2(\mathbf{X}^T \mathbf{X})^{-1}$.

Does that hold true when interaction effects are included, that is $\mathbf{X}$ contains variables and the products of each other?