Step-by-step Solution:

Given the matrix: \[ A = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \] We need to find \( A^2 \). Step 1: Compute \( A^2 \) \[ A^2 = A \times A = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \times \begin{bmatrix} a & b \\ b & a \end{bmatrix} \] Using matrix multiplication: \[ A^2 = \begin{bmatrix} a \cdot a + b \cdot b & a \cdot b + b \cdot a \\ b \cdot a + a \cdot b & b \cdot b + a \cdot a \end{bmatrix} \] \[ A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \\ 2ab & a^2 + b^2 \end{bmatrix} \] Comparing with the given form: \[ A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix} \] We conclude: \[ \alpha = a^2 + b^2, \quad \beta = 2ab \] Thus, the correct option is: \[ \text{Option B: } \alpha = a^2 + b^2, \quad \beta = 2ab \]