If $I_3$ is the identity matrix of order $3$, then $I_3^{-1}$ is:

By definition, the identity matrix $I$ is a square matrix such that for any square matrix $A$ of the same order, $AI=IA=A$.For an inverse to exist, a matrix must be non-singular ($|A|\ne 0$).For the identity matrix $I_3$:
$$|I_3|=1\ne 0$$
The property of the identity matrix is that $I\cdot I=I$.Comparing this with the definition of the inverse ($A\cdot A^{-1}=I$), we can see that:
$$I_3^{-1}=I_3$$
Alternatively, $\text{adj}(I_3)=I_3$ and $I_3^{-1}=\frac{\text{adj}(I_3)}{|I_3|}=\frac{I_3}{1}=I_3$.