$A = \begin{bmatrix} \alpha &\beta \\ \beta & \alpha \end{bmatrix}$
$A^{2} = \begin{bmatrix} \alpha &\beta \\ \beta & \alpha \end{bmatrix} = \begin{bmatrix} \alpha & \beta\\ \beta & \alpha \end{bmatrix} \begin{bmatrix} \alpha & \beta\\ \beta & \alpha \end{bmatrix} = \begin{bmatrix} \alpha^{2} + \beta^{2} & \alpha\beta + \alpha\beta\\ \alpha\beta + \alpha\beta & \alpha^{2} + \beta^{2} \end{bmatrix}$
$A^{2} = \begin{bmatrix} \alpha^{2} + \beta^{2} & 2\alpha\beta\\ 2\alpha\beta & \alpha^{2} + \beta^{2} \end{bmatrix}$ = $\begin{bmatrix} a & b\\ c & d \end{bmatrix}$
Hence,
$a + c = \alpha ^{2} + \beta^{2}+ 2\alpha\beta$
$a + c = (\alpha + \beta)^{2}$
Answer is option A