Solution of a matrix problem on a mathematical safe for locks of the same kind. I (Q2577247)

A mathematical safe is given as \quad a system \quad \({\mathbf S}({\mathbf Z}, {\mathbf B}, \langle {\mathbf Z} \rangle )\), \quad where \quad \({\mathbf Z} = \{ z_1, z_2, \dots, z_N\}\) is a set of locks, \({\mathbf B} = \{ {\mathbf b} = (b_1, b_2, \dots, b_N)\}\) is the set of states of the safe, \(b_l \in \{0,1, \dots, k_l - 1 \}\) is the state of the \(l\)th lock, and \( \langle {\mathbf Z} \rangle = \{{\mathbf Z}_1,{\mathbf Z}_2, \dots, {\mathbf Z}_N \}\), \({\mathbf Z}_l \in 2^{\mathbf Z}\), \(z_j \in {\mathbf Z}_l \,(l = 1, 2, \dots, N)\). Here, the safe in which all the locks are located in the form of a rectangular matrix are considered and by \({\mathbf Z}_l\) are denoted all the locks that belong to the same type, i. e., \(k_l = K\). The solution is searched for in the form of a matrix by which elements \(b_{i,j}\) are transformed into zero. The presented solution is discussed with regard to the two necessary conditions of solvability of the system and illustrated by several examples.