> [!Definition] Definition (Identity matrix) > Let $n\in \mathbb{N}^{+}.$ The (real) identity matrix of order $n$ is the [[Real Matrices|real matrix]] $I_{n}\in\text{Mat}_{nn}(\mathbb{R})$ whose $(i,j)$ entry is given by $\delta_{ij}$ for $1\leq i,j\leq n$, where $\delta_{ij}$ denotes the [[Kronecker Delta Function]]. # Properties By [[Product with Real Identity Matrix]], for all $A\in \text{Mat}_{mn}(\mathbb{R}),$ $AI_{n}=I_{m}A=A.$