In this paper we give the matrix theory of some regression credibility models and we try to demonstrate what kind of data is needed to apply linear algebra in the regression credibility models. Just like in the case of classical credibility model we will obtain a credibility solution in the form of a linear combination of the individual estimate (based on the data of a particular state) and the collective estimate (based on aggregate USA data). To illustrate the solution with the properties mentioned above, we shall need the well-known representation formula of the inverse for a special class of matrices. To be able to use the better linear credibility results obtained in this study, we will provide useful estimators for the structure parameters, using the matrix theory, the scalar product of two vectors, the norm and the concept of perpendicularity with respect to a positive definite matrix given in advance, an extension of Pythagoras' theorem, properties of the trace for a square matrix, complicated mathematical properties of conditional expectations and of conditional covariances.
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