If \(A = \begin{bmatrix}a&b\\b&a\end{bmatrix}\) and \(A^2 = \begin{bmatrix}\alpha&\beta\\\beta&\alpha\end{bmatrix}\), then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\alpha = a^2 + b^2,\ \beta = 2ab\)
Multiplying row by column:
\( \displaystyle A^2 = \begin{bmatrix}a&b\\b&a\end{bmatrix}\begin{bmatrix}a&b\\b&a\end{bmatrix} \)
\( \displaystyle A^2 = \begin{bmatrix}a^2 + b^2&ab + ba\\ba + ab&b^2 + a^2\end{bmatrix} = \begin{bmatrix}a^2 + b^2&2ab\\2ab&a^2 + b^2\end{bmatrix} \)
Comparing with \(A^2\) as given,
\(\alpha = a^2 + b^2\) and \(\beta = 2ab\)