> [!NOTE] Lemma (Product with Real Identity Matrix) > Let $A\in \text{M}_{mn}$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Then the [[Real Matrix Product|product]] $A I_{n}=I_{m}A=A$ where $I_{n}$ denotes the [[Real Identity Matrix|identity matrix]] of order $n.$ **Proof**: We have $\begin{align} \text{the $(i, j)$th entry of $A I_n$} &= \sum_{k=1}^n a_{i k} \delta_{k j}=a_{i j}\\\text{the $(i, j)$th entry of $I_m A$}&=\sum_{k=1}^m \delta_{i k} a_{k j}=a_{i j} \end{align}$