Given a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), find the identity matrix \( I \) such that \( AI = IA = A \).

Step 1 ：Step 1: The identity matrix for any square matrix, including a 2x2 matrix, is a matrix where all the elements in the main diagonal are 1 and all other elements are 0. Therefore, the 2x2 identity matrix is \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \).

Step 2 ：Step 2: To verify that \( AI = IA = A \), we multiply the given matrix \( A \) by the identity matrix \( I \) from both the left (to get \( AI \)) and the right (to get \( IA \)).

Step 3 ：Step 3: \( AI = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

Step 4 ：Step 4: \( IA = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

Step 5 ：Step 5: We can see that in both cases, the result is the original matrix \( A \). Therefore, \( AI = IA = A \), which confirms that \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) is indeed the identity matrix for \( A \).