The authors study structured matrices which consist of a band part and quasiseparable parts above and below the band, extending algorithms known for quasiseparable matrices to a wider class of matrices. The first section represents the introduction. The second section gives the definitions and auxiliary relations for the band part of the matrix. In the third section one derives some relations for columns and rows of out-of-band quasiseparable matrices which are used in the sequel. The fourth section focuses on the notion of rank numbers and minimal generators for a block matrix, obtaining that for every matrix the minimal generators exist. The authors show the connection of minimal generators with rank numbers. The fifth section derives a fast algorithm for the multiplication of a quasiseparable matrix by a vector. A similar algorithm is presented in the sixth section. This one is used to obtain the solution of a system having a triangular out-of-band quasiseparable matrix. The seventh section contains the main results of the paper on the LDU factorization for an out-of-band quasiseparable matrix. The results of section seven are summarized to obtain a fast algorithm for solution of a system of linear alegbraic equations. The authors also analyse the complexity of this algorithm. The results of computer experiments with the linear solver obtained in this paper and some other algorithms are presented in the ninth section.