For determining the Inverse of an identity matrix, let us take** I**_{2} as a given 2x2 Identity matrix.

So, I_{2} = $\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** I**_{2}.

C = $\begin{bmatrix}

1 &0 \\

0 & 1

\end{bmatrix}$

adj I_{2} = $C^T$ = $\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.