A matrix that consists of ones as diagonal elements and zeros as non-diagonal elements is called an identity matrix.
In a special case, each element in the main diagonal (or leading diagonal) can be one and the remaining non-diagonal entries can be zeros in a matrix.
$I$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \color{red} \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix}$
There are two popular meanings for the word “identity” in English language.
Hence, this type of matrix is called an identity matrix, and simply denoted by $I$ in mathematics. It is also called a unit matrix.
An identity matrix is basically a square matrix.
Due to these two reasons, a unit matrix is a diagonal matrix principally and also a scalar matrix.
The following three examples help you to understand how to express the identity matrices of different orders.
$I_2$ $\,=\,$ $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $
It is a matrix of the order $2 \times 2$ but it is an identity matrix and also a square matrix. Hence, it is known as an identity matrix of order $2$. It is simply denoted by $I_2$ in matrix.
$I_3$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix} $
It is a matrix of the order $3 \times 3$. It is an identity matrix and also a square matrix. Therefore, it is called an identity matrix of order $3$ and simply denoted by $I_3$ in mathematics.
$I_4$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $
It is a matrix of the order $4 \times 4$. It is an identity matrix and also a square matrix. So, it is called an identity matrix of order $4$ and simply denoted by $I_4$ in mathematics.
In this way, an identity matrix of any order can be expressed in mathematics.
$I_n$ $\,=\,$ $\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \color{red} \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix}$
If the number of entries in each row is $n$ and the number of elements in each column is $n$, then it is called a matrix of the order $n \times n$. It is an identity matrix and also a square matrix. Hence, it is called an identity matrix of order $n$ and simply denoted by $I_n$ in mathematics.
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