Question
If \[A = \left[ \begin{array}{l}
a\,\,\,\,\,\,\,\,b\\
b\,\,\,\,\,\,\,\,a
\end{array} \right]{\rm{and }}\,\,{A^2} = \left[ \begin{array}{l}
\alpha \,\,\,\,\,\,\beta \\
\beta \,\,\,\,\,\,\alpha
\end{array} \right],\] 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 :
$$\alpha = {a^2} + {b^2},\beta = 2ab$$
Solution :
\[{A^2} = \left[ \begin{array}{l}
\alpha \,\,\,\,\,\,\beta \\
\beta \,\,\,\,\,\,\alpha
\end{array} \right] = \left[ \begin{array}{l}
a\,\,\,\,\,\,b\\
b\,\,\,\,\,\,a
\end{array} \right]\left[ \begin{array}{l}
a\,\,\,\,\,\,\,b\\
b\,\,\,\,\,\,\,a
\end{array} \right]\]
\[\,\,\,\,\,\,\,\,\, = \left[ \begin{array}{l}
{a^2} + {b^2}\,\,\,\,\,\,\,\,2ab\\
\,\,\,2ab\,\,\,\,\,\,\,\,\,\,\,{a^2} + {b^2}
\end{array} \right]\]
$$\alpha = {a^2} + {b^2};\beta = 2ab$$