Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 1: JEE Advanced Paper 1 - 2020

Let $P_1=I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right], P_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right], P_3=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right],$$P_4=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right], P_5=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],P_6=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],$ and
$X=\sum _{k=1}^6{P}_K\left[\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 3& 2& 1\end{array}\right]P_K^T$
where $P_K^T$ denotes the transpose of the matrix $P_K.$ Then which of the following options is/are correct?

Let $x\in R$ and let $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right], Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}.$
Then which of the following options is/are correct?

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let

$E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$

If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identity matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE ?

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

Let $\mathrm{M}=\left[\begin{array}{cc}{\sin }^4\mathrm{\theta }& -1-{\sin }^2\mathrm{\theta }\\ 1+{\cos }^2\mathrm{\theta }& {\cos }^4\mathrm{\theta }\end{array}\right]=\mathrm{\alpha }\mathrm{I}+\mathrm{\beta }{\mathrm{M}}^{-1},$ where $\mathrm{\alpha }=\mathrm{\alpha }\left(\mathrm{\theta }\right)$ and $\mathrm{\beta }=\mathrm{\beta }\left(\mathrm{\theta }\right)$ are real number, and $\mathrm{I}$ is the $2\times 2$ identity matrix. If ${\mathrm{\alpha }}^{\mathrm{*}}$ is the minimum of the set { $\mathrm{\alpha }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\}$ and ${\mathrm{\beta }}^{\mathrm{*}}$ is the minimum of the set $\left\{\mathrm{\beta }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\right\},$ then the value of ${\mathrm{\alpha }}^{\mathrm{*}}+{\mathrm{\beta }}^{\mathrm{*}}$ is

Let $M=\left[\begin{array}{ccc}0& 1& a\\ 1& 2& 3\\ 3& b& 1\end{array}\right]$ and $\mathrm{a}\mathrm{d}\mathrm{j}M=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?

Let $\mathrm{S}$ be the sample space of all $3\times 3$ matrices with entries from the set $\left\{0,\mathrm{ }1\right\}.$ Let the events $E_1$ and $E_2$ be given by $E_1=\left\{A\in S:detA=0\right\}$ and $E_2=\{A\in S:$ sum of entries of $\mathrm{A}$ is $7\}.$ If a matrix is chosen at random from $\mathrm{S},$ then the conditional probability $P\left(E_1|E_2\right)$ equals____