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If \( \mathrm{A} \& \mathrm{~B} \) are square matrices of order 2 such that \( \mathrm{A}+\operatorname{adj}\left(\mathrm{B}^{\mathrm{T}}\right)=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right] \& \mathrm{~A}^{\mathrm{T}}-\operatorname{adj}(\mathrm{B})=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] \), then-
(A) \( \mathrm{B} \) is symmetric matrix
(B) \( \mathrm{A}^{\mathrm{n}}=\mathrm{A} \forall \mathrm{n} \in \mathrm{N} \)
(C) \( \left|\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\mathrm{A}^{4}+\mathrm{A}^{5}\right|=0 \)
(D) \( \left|B+B^{2}+B^{3}+B^{4}+B^{5}\right|=0 \)
\( \mathrm{P} \)
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