Question

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

• 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. $\alpha = a^2 + b^2 , \beta = a^2 - b^2$

Answer 1

Answer: (C)

$\alpha = a^2 + b^2 , \beta = 2ab$

Answer 2

$A^{2}\begin{bmatrix}\alpha&\beta \\\ \beta&\alpha\end{bmatrix}$ $\Rightarrow\begin{bmatrix}\alpha&\beta\\\ \beta&\alpha\end{bmatrix} = \begin{bmatrix}a&b\\\ b&a\end{bmatrix}\begin{bmatrix}a&b\\\ b&a\end{bmatrix}$ $\Rightarrow \begin{bmatrix}\alpha&\beta\\\ \beta&\alpha\end{bmatrix} = \begin{bmatrix}a^{2} +b^{2}&2ab\\\ 2ab&a^{2}+b^{2}\end{bmatrix}$ $\Rightarrow \alpha = a^{2}+b^{2}$ and $2ab = \beta$