Identity matrix - Aetherwisp

The identity matrix $\mathrm{I}_{n}$ of size $n$ is an $n\times n$ square matrix with ones on the diagonal and zeros elsewhere:

\mathrm{I}_{n}=\begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 \end{pmatrix}

It can be defined by its entries using the Kronecker delta as

(\mathrm{I}_{n})_{ij}=\delta_{ij}

Given an $n\times m$ matrix $\mathrm{A}$ , matrix multiplication of $\mathrm{A}$ with the identity matrix does not result in any change: $\mathrm{I}_{n}\mathrm{A}=\mathrm{A}\mathrm{I}_{m}=\mathrm{A}$ .
The rank of the identity matrix is its size: $\text{rank }\mathrm{I}_{n}=n$ .
The trace is also equal to its size: $\text{Tr }\mathrm{I}_{n}=n$ .
The determinant is one regardless of size: $\det \mathrm{I}_{n}=1$ .