Trace of product of linear transformations is invariant under cyclic permutations

Statement
Suppose $$V$$ is a finite-dimensional vector space over a field $$F$$. Consider linear transformations $$A_1,A_2,\dots,A_n \in \operatorname{End}(V)$$. Then, for any $$1 \le i \le n$$, the trace of the linear transformation $$A_iA_{i+1}\dots A_nA_1A_2 \dots A_i$$ is independent of the choice of $$i$$.

For instance, when $$n = 3$$, this states that $$A_1A_2A_3$$, $$A_2A_3A_1$$, and $$A_3A_1A_2$$ all have the same trace.

Related facts

 * Trace of product of linear transformations is not invariant under arbitrary permutations

Facts used

 * 1) uses::Trace of product of two linear transformations is independent of their order