Question

Consider the group \( G = \{ A \in M_2(\mathbb{R}) : AA^T = I_2 \ \) with respect to matrix multiplication. Let \[ Z(G) = \{ A \in G : AB = BA, \, for all \, B \in G \}. \] Then, the cardinality of \( Z(G) \) is:}

• A. 1
• B. 2
• C. 4
• D. Infinite

Hint:

The centralizer of the orthogonal group \( O(2) \) in \( M_2(\mathbb{R}) \) is limited to the identity and its negative, because these are the only matrices that commute with all other matrices in \( O(2) \).

Answer 1

The Correct Option is B

The group \( G \) consists of the orthogonal matrices in \( M_2(\mathbb{R}) \), i.e., matrices \( A \) such that \( A A^T = I_2 \), where \( I_2 \) is the identity matrix. These matrices represent the orthogonal group \( O(2) \), which consists of rotations and reflections in the plane. The centralizer \( Z(G) \) consists of those matrices \( A \in G \) that commute with every element \( B \in G \). For \( 2 \times 2 \) orthogonal matrices, the only matrices that commute with every other matrix in \( O(2) \) are the scalar multiples of the identity matrix, i.e., \( A = \pm I_2 \), where \( I_2 \) is the identity matrix. Thus, \( Z(G) \) contains exactly two elements: the identity matrix \( I_2 \) and the negative identity matrix \( -I_2 \). Therefore, the cardinality of \( Z(G) \) is 2.