On a switch-setting problem arising from a variation of the `Lights Out' game (Q2883286)

Consider two \(m\times n\) matrices whose entries come from residue classes mod \(k\). We are allowed to add 1 to every element of a row and also to add 1 to every element of a column. The paper characterizes when can we reach from the first matrix the second with a sequence of such operations. The problem and its solution is extended to \(\ell\)-dimensional arrays, where the entries can be elements of an Abelian group. The allowed operation is fixing all but one coordinates to obtain a column in one of the \(\ell\) directions and add the same element of the Abelian group to every entry in this column. While this operation is repeated, a different element of the Abelian group may be used.