Let P _ 4 = [ array lll 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 array ], P _ 5 = [ array lll 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 array ], P _ 6 = [ array lll 0 & 0 & 1 0 & 1 & 0 1 & 0 & 0 array ], Where P k T denotes the transpose of matrix P k . Then which of the following options is/are correct?

Let P _ 4 = [ array lll 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 array ], P _ 5 = [ array lll 0 & 0…

Mathematics · JEE Advanced · NTA Exams — Matrices and Determinants

Let

\[\boldsymbol{P}_{4}=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right], \boldsymbol{P}_{5}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right], \boldsymbol{P}_{6}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right],\]

Where P_k^T denotes the transpose of matrix P_k. Then which of the following options is/are correct?

X is a symmetric matrix
The sum of diagonal entries of X is 18 .
X – 30I is an invertible matrix.

Answer
(D) X – 30 I is an invertible matrix.

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