Answer
$ I_2 A=AI_2=A$ so, $I_2$ is identity for $2 \times 2$ matrices.
Work Step by Step
$AI_2=\begin{bmatrix} a & b \\ c&d\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0& 1\end{bmatrix} \\=\begin{bmatrix} a+0 & 0+b \\c+0 & 0+d \end{bmatrix} \\=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
Next, $I_2 A= \begin{bmatrix} 1 & 0 \\ 0& 1\end{bmatrix} \begin{bmatrix} a & b \\ c&d\end{bmatrix} \\=\begin{bmatrix} a+0 &b+0 \\0+c & 0+d \end{bmatrix} \\=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
This has been verified that $ I_2 A=AI_2=A$ so, $I_2$ is identity for $2 \times 2$ matrices.
