Question

Let $P_{1}=I=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]$, $P_{2}=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0\end{array}\right]$, $P_{3}=\left[\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1\end{array}\right]$, $P_{4}=\left[\begin{array}{lll}0 & 1 & 0 \ 0 & 0 & 1 \ 1 & 0 & 0\end{array}\right]$, $P_{5}=\left[\begin{array}{lll}0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0\end{array}\right]$, $P_{6}=\left[\begin{array}{lll}0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0\end{array}\right]$ and $X=\sum_{k=1}^{6} P_{K}\left[\begin{array}{lll}2 & 1 & 3 \ 1 & 0 & 2 \ 3 & 2 & 1\end{array}\right] P_{K}^{T}$ where $\mathrm{P}_{\mathrm{K}}^{\mathrm{T}}$ denotes the transpose of the matrix $\mathrm{P}_{\mathrm{K}}$. Then which of the following options is/are correct?

Accepted Answer

(B) The sum of diagonal entries of X is 18; (C) If $X\left[\begin{array}{l}1 \ 1 \ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \ 1 \ 1\end{array}\right]$, then $\alpha=30$; (D) X is a symmetric matrix

Answer

(A) X - 30I is an invertible matrix

JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Matrices

Solution

More JEE Advanced questions on Matrices