The sum of orthogonal matrices in Mn(Zk)

We show that every A∈Mn(Z2k-1) can be written as a sum of orthogonal matrices (QQT=QTQ=I) in Mn(Z2k-1). Moreover, we show that every A∈Mn(Z2k) can be written as a sum of orthogonal matrices in Mn(Z2k) if and only if the row sums and column sums of A have the same parities.