Dimension reduction for regression analysis has been one of the most popular topics in the past two decades. The reason is that if we can describe the conditional distribution of the response variable Y given the covariate vector X through only a few linear combinations of X, then the effective dimension of the regression model will be dramatically reduced. This is also known as sufficient dimension reduction (SDR) in the literature. The study in this area sees much progress with the introduction of the inverse regression method pioneered by Li (1991). Most of these methods are centered around a matrix, called the central matrix, which is then used to estimate the central subspace, spanned by the weight vectors in the linear combinations of the covariate vector X. In this work, we propose a new method to construct a central matrix from existing ones. In the process we propose a test for the evenness and oddness of the regression function and a method to combine standardized non-zero central matrices. Another key issue in SDR is the estimation of the dimension. Based on the newly constructed central matrix above, we introduce a BIC (Bayesian Information Criterion) method to estimate the dimension of the central subspace. In the past few years semi-parametric methods have brought much development into the field in term of finding an efficient estimator. Here we extend the semi-parametric methods for efficient estimator to the case with linearity condition for the central subspace model and the central mean subspace model.