Trace of Alternating Product of Matrices and Almost Zero Matrices

Theorem
Let $R$ be a ring with unity.

Let $n, m$ be positive integers.

Let $E_{ij}$ denote the $n \times n$ matrix with only zeroes except a $1$ at the $\tuple {i, j}$th element.

Let $A_1, \ldots, A_m \in R^{n \times n}$.

Let $i_k, j_k \in \set {1, \ldots, n}$ for $k \in \set {1, \ldots, m}$.

Let $i_0 = i_m$ and $j_0 = j_m$.

Then:
 * $\map \tr {A_1 E_{i_1, j_1} A_2 E_{i_2, j_2} \cdots A_m E_{i_m, j_m} } = \ds \prod_{k \mathop = 1}^m \sqbrk {A_k}_{j_{k - 1} i_k}$

Proof
Use induction and the facts $E_{i j} A E_{k l} = A_{j k} E_{i l}$ and $\map \tr {A E_{i j} } = A_{j i}$ (induction basis).