Two new algorithms are presented for determining the elements of the matrix function Al of a digital system of which the matrix of dynamics has nonzero elements only in its superdiagonal (ones) and in its last row (a particular combination of the coefficients of the transfer function denominator). Advantages and disadvantages of these algorithms are discussed. It is shown that when using the matrix function Al only for the calculation of the system response, the second algorithm presented here seems to be particulary useful due to the simplicity of its application and due to the fact that it allows us to omit the calculations of the eigenvalues of the matrix of dynamics, which seem to have been necessary for most of the known methods. It also reduces the number of computer operations which must be performed from n3(n-2) using direct matrix multiplication, to n(n-2). The second algorithm can be efficiently used for the determination of the elements of limited power of the matrix of dynamics. Both of the algorithms presented in this paper are particularly useful when applied to the matrix of dynamics of a digital filter, of which the transfer function is presented in the direct form of realization, since, the matrix of dynamics of such a digital system is expressed directly in a proper form.