The Identity Matrix is a Two-Sided Multiplicative Identity

Let $m,n\in$ $\mathbb{N}$ and let $A$ be a real $m\times n$ matrix. Let $I_m$ and $I_n$ denote the identity matrices of sizes $m$ and $n$, and let the products be taken in the sense of Product of Real Matrices. Then

$$A\,I_n=A\qquad\text{and}\qquad I_m\,A=A.$$

In particular, for a square matrix $A$ of size $n$, $A\,I_n=I_n\,A=A$.