JEE (Advanced) 2019 Paper-2 Question 37

Question 37

Let
$$P_1 = I = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}, P_2 = \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1\\0 & 1 & 0\end{bmatrix}, P_3 = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0\\0 & 0 & 1\end{bmatrix},$$
$$P_4 = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1\\1 & 0 & 0\end{bmatrix}, P_5 = \begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0\\0 & 1 & 0\end{bmatrix}, P_6 = \begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0\\1 & 0 & 0\end{bmatrix},$$
and $$X = \sum_{k = 1}^6 P_k \begin{bmatrix}2 & 1 & 3 \\1 & 0 & 2\\3 & 2 & 1\end{bmatrix} P_k^T$$
where $$P_k^T$$ denotes the transpose of the matrix $$P_k$$. Then which of the following options is/are correct?



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