Let \(\mathbf{I}_n\) be the \(n \times n\) matrix defined by,

\[\mathbf{I}_n = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}\] \[\mathbf{I}_n = \operatorname{diag}(1, 1, \dots, 1)\]

The matrix \(\mathbf{I}_n\) is called the identity matrix of order \(n\). The identity matrix has the property that for any \(n \times n\) matrix \(\mathbf{A}\), we have,

\[\mathbf{I}_n \mathbf{A} = \mathbf{A} \mathbf{I}_n = \mathbf{A}\]