Define a relation R over a class of n times n | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) relation $R$ is defined by $A R B \iff B = P A P^{-1}$ for some non-singular matrix $P$.$1$. Reflexive: For any matrix $A$,we can choose $P = I$ (

Vedclass · Accepted Answer

$R$ is an equivalence relation.

Vedclass · Answer

(C) relation $R$ is defined by $A R B \iff B = P A P^{-1}$ for some non-singular matrix $P$.$1$. Reflexive: For any matrix $A$,we can choose $P = I$ (the identity matrix). Then $I A I^{-1} = I A I = A$. Thus,$A R A$ holds. So,$R$ is reflexive.$2$. Symmetric: Suppose $A R B$. Then $B = P A P^{-1}$ for some non-singular matrix $P$. We can write $A = P^{-1} B P$. Let $Q = P^{-1}$. Since $P$ is non-singular,$Q$ is also non-singular. Then $A = Q B Q^{-1}$. Thus,$B R A$ holds. So,$R$ is symmetric.$3$. Transitive: Suppose $A R B$ and $B R C$. Then $B = P A P^{-1}$ and $C = Q B Q^{-1}$ for some non-singular matrices $P$ and $Q$. Substituting $B$ in the second equation,we get $C = Q (P A P^{-1}) Q^{-1} = (Q P) A (P^{-1} Q^{-1}) = (Q P) A (Q P)^{-1}$. Since the product of two non-singular matrices $Q P$ is also non-singular,$A R C$ holds. So,$R$ is transitive.Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as $A R B$ if and only if there exists a non-singular matrix $P$ such that $P A P^{-1} = B$. Then which of the following is true?

Similar Questions

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R = \{(L_{1}, L_{2}) : L_{1} \text{ is perpendicular to } L_{2}\}$. Show that $R$ is symmetric but neither reflexive nor transitive.

Let the relation $R_{1}$ be defined on $R$ as $a R_{1} b$ if $1+ab > 0$. Then

The relation $R$ defined in the set of natural numbers $N$ as $aRb \iff b$ is divisible by $a$ is:

Let $A = \{2, 3, 6, 7\}$ and $B = \{4, 5, 6, 8\}$. Let $R$ be a relation defined on $A \times B$ by $(a_1, b_1) R (a_2, b_2)$ if and only if $a_1 + a_2 = b_1 + b_2$. Then the number of elements in $R$ is ...........

Let $R$ be the relation on the set $\mathbb{R}$ of all real numbers defined by $a \ R \ b$ if $|a - b| \le 1$. Then $R$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module