We establish connection between product of two matrices of order $k\times k$ over a field and the product of the k-mappings corresponding to the $k$-operations, defined by these matrices. It is proved that, in contrast to the binary case, for arity $k\geq 3$ the components of the $k$-permutation inverse to a $k$-permutation, all components of which are polynomial $k$-quasigroups, are not necessarily $k$-quasigroups although are invertible at least in two places. Some transformations with the help of permutations of orthogonal systems of polynomial $k$-operations over a field are considered.
"Polynomial $k$-ary operations, matrices, and $k$-mappings." J. Gen. Lie Theory Appl. 4 1 - 8, 2010. https://doi.org/10.4303/jglta/G100301