We consider the following extremal problem: Among all matrices that map a given point x of the Euclidean spaceRnto a given point b of the Euclidean spaceRm, find a matrix of minimal norm. The article uses the Hölder norm of vectors and matrices depending on a parameter p. It is shown that for every p∈(1,+∞) there exists a unique solution of the stated problem and an explicit formula for it is derived. Limits of the optimal matrix are found as p approaches the boundary values p→1 and p→+∞. It is established that the limiting matrices are solutions of the corresponding limiting extremal problems. However, unlike the case p∈(1,+∞), uniqueness of these limiting solutions is not guaranteed. A full description of the entire set of solutions of the nonsmooth limiting problems is given. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.