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# If $A=\begin{bmatrix}a&b\\b&a\end{bmatrix}$ and $A^2=\begin{bmatrix}\alpha&\beta\\\beta&\alpha\end{bmatrix}$ then

$\begin{array}{1 1}(a)\;\alpha=2ab,\beta=a^2+b^2\\(b)\;\alpha=a^2+b^2,\beta=ab\\(c)\;\alpha=a^2+b^2,\beta=2ab\\(d)\;\alpha=a^2+b^2,\beta=a^2-b^2\end{array}$

Given :
$A^2=\begin{bmatrix}\alpha&\beta\\\beta&\alpha\end{bmatrix}$
$A=\begin{bmatrix}a&b\\b&a\end{bmatrix}$
$A^2=\begin{bmatrix}a&b\\b&a\end{bmatrix}\begin{bmatrix}a &b\\b&a\end{bmatrix}$
$\Rightarrow \begin{bmatrix}a^2+b^2&2ab\\2ab&a^2+b^2\end{bmatrix}$
$\Rightarrow \alpha=a^2+b^2,\beta=2ab$
Hence (c) is the correct answer.