What is the inverse matrix of matrix pmatrix 1 & 0 \\ 0 & 1 pmatrix ? Write it.

The given matrix is the identity matrix of order $2\times 2$, denoted as $I_2$:

$$I_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

Step-by-Step Solution:

1. Definition of Inverse Matrix: The inverse of a matrix $A$, denoted as $A^{-1}$, satisfies the property: $$A\cdot A^{-1}=A^{-1}\cdot A=I$$ where $I$ is the identity matrix.

2. Inverse of the Identity Matrix: Since $I_2$ already satisfies $I_2\cdot I_2=I_2$, its inverse is itself.

3. Verification: $$I_2\cdot I_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I_2$$

Thus, the inverse of the given matrix is:

$$\boxed{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)}$$