If A=abba and A2=αββα] , then

A2=[αββα]=[abba][abba] =[a2+b22ab2aba2+b2] ∴α=a2+b2,β=2ab

If A=abba and A2=αββα] , then

A2=[αββα]=[abba][abba] =[a2+b22ab2aba2+b2] ∴α=a2+b2,β=2ab

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If $A = [\begin{matrix} a & b \\ b & a \end{matrix}]$ and $A^{2} = [\begin{matrix} α & β \\ β & α \end{matrix}]]$ , then

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$α = a^{2} + b^{2}, β = a b$

$α = a^{2} + b^{2}, β = 2 a b$

$α = a^{2} + b^{2}, β = a^{2} - b^{2}$

$α = 2 a b, β = a^{2} + b^{2}$

answer is B.

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Detailed Solution

$A^{2} = [\begin{matrix} α & β \\ β & α \end{matrix}] = [\begin{matrix} a & b \\ b & a \end{matrix}] [\begin{matrix} a & b \\ b & a \end{matrix}] = [\begin{matrix} a^{2} + b^{2} & 2 a b \\ 2 a b & a^{2} + b^{2} \end{matrix}] ∴ α = a^{2} + b^{2}, β = 2 a b$