An Identity matrix is a type of diagonal matrix in which all the diagonal elements are equal to 1 and 0 elsewhere. An Identity matrix is denoted by In. In algebra, 1 is the multiplicative identity (1 . a = a = a . 1). So, in matrices, In is the multiplicative identity.
In A = A = A In.

I2 = $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$, I3 = $\begin{bmatrix}
1 &0 &0 \\
0 &1 &0 \\
0 &0 &1
\end{bmatrix}$


The determinant of an identity matrix of any order is equal to 1.


To prove this, let us take a 3x3 identity matrix.
In = $\begin{bmatrix}
1 &0 &0 \\
0 &1 &0 \\
0 &0 &1
\end{bmatrix}$

Now, we have to calculate $\left | I_{n} \right |$ for which we use row expansion method.
$\left | I_{n} \right |$ = 1 $\begin{vmatrix}
1 &0 \\
0 &1
\end{vmatrix}$ - 0 $\begin{vmatrix}
0 &0 \\
0 &1
\end{vmatrix}$ + 0 $\begin{vmatrix}
0 &1 \\
0 &0
\end{vmatrix}$

= 1(1 - 0) - 0(0 - 0) + 0(0 - 0)
= 1
$\left | I_{n} \right |$= 1

For determining the Inverse of an identity matrix, let us take I2 as a given 2x2 Identity matrix.
So, I2 = $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$

If A is the matrix, then, its inverse is determined using the below formula:
A-1 = $\frac{adj A}{\left | A \right |}$
Then, $\left | I_{2} \right |$ = 1

For determining the adj A, lets first find cofactor matrix of I2.
C = $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$

adj I2 = $C^T$ = $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$

Hence, inverse of an identity matrix is the matrix itself.

$I_{2}^{-1}$ = 1. $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$

= $\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}$

So, we can say that $ I_{2} = I_{2}^{-1}$

It is clear that the determinant of an identity matrix of any order is always equal to 1 and the inverse of an identity matrix is the matrix itself. This is one of the identity matrix properties.