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 $(i,j)$th position.

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

Let $i_k,j_k\in\{1,\ldots,n\}$ for $k\in\{1,\ldots,m\}$, and define $i_0=i_m$, $j_0=j_m$.

Then $\operatorname{tr}\left(A_1 E_{i_1,j_1} A_2 E_{i_2,j_2} \cdots A_{m} E_{i_m,j_m}\right) = \displaystyle \prod_{k=1}^m (A_k)_{j_{k-1}i_k}$.

Proof
Use induction and the facts $E_{ij} A E_{kl} = A_{jk} E_{il}$ and $\operatorname{tr}(A E_{ij}) = A_{ji}$ (induction basis).