In this paper, the concept of the basis matrix of B-splines is presented. A general matrix representation, which results in an explicitly recursive matrix formula, for nonuniform Bspline curves of an arbitrary degree is also presented by means of the Toeplitz matrix. New recursive matrix representations for uniform B-spline curves and Bézier curves of an arbitrary degree are obtained as special cases of that for nonuniformB-spline curves. The recursive formula for the basis matrix can be substituted for de Boor-Cox’s formula for B-splines, and it has a better time complexity than de Boor-Cox’s formula when used for computation and conversion of Bspline curves and surfaces between different CAD systems. Finally, some applications of the matrix representations are given in the paper.