📂Matrix Algebra

Identity Matrix, Unit Matrix

Definition

A diagonal matrix of size $n\times n$ with all diagonal elements being $1$ is called an identity matrix or unit matrix, denoted as $I_{n}$ or $I_{n\times n}$.

$$I_{n\times n}= \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}$$

Description

The identity matrix is the identity element for matrix multiplication. This means that for any $n\times n$ matrix $A$, the following equation holds:

$$I_{n}A=A=AI_{n}$$

Properties

Determinant

Since the identity matrix is a diagonal matrix, its determinant is $1$.

$$\det I = 1$$