Exercise 24, § 8

Let $X_i$ ($1 \leq i \leq m$) be square matrices of the same order $n$ over the commutative ring A. Show that if $m$ is even then (Exercise 23)

and if $m$ is odd

(If $\lambda$ is the cyclic permutation $(1, 2, \ldots, m)$, $C$ the cyclic subgroup of $S_m$ generated by $\lambda$ and $H$ the subgroup of $S_m$ consisting of the permutations leaving $m$ invariant, note that every permutation in $S_m$ can be written uniquely as $\sigma \tau$ with $\sigma \in H$ and $\tau \in C$; then use the fact that the trace of a product of matrices is invariant under cyclic permutation of the factors.)