In the set of all 3 times 3 real matrices,a r | vedclass

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

Question

(D) Let the relation be defined as $R = \{(A, B) \mid B = P^{-1} A P 	ext{ for some non-singular matrix } P\}$.For reflexivity: Since $A = I^{-1} A I

Vedclass · Accepted Answer

an equivalence relation

Vedclass · Answer

(D) Let the relation be defined as $R = \{(A, B) \mid B = P^{-1} A P 	ext{ for some non-singular matrix } P\}$.For reflexivity: Since $A = I^{-1} A I$ where $I$ is the identity matrix,$(A, A) \in R$. Thus,$R$ is reflexive.For symmetry: Let $(A, B) \in R$. Then $B = P^{-1} A P$. Multiplying by $P$ on the left and $P^{-1}$ on the right,we get $P B P^{-1} = A$. Let $Q = P^{-1}$. Then $A = Q^{-1} B Q$. Thus,$(B, A) \in R$. So,$R$ is symmetric.For transitivity: Let $(A, B) \in R$ and $(B, C) \in R$. Then $B = P^{-1} A P$ and $C = Q^{-1} B Q$ for some non-singular matrices $P$ and $Q$. Substituting $B$,we get $C = Q^{-1} (P^{-1} A P) Q = (P Q)^{-1} A (P Q)$. Since $PQ$ is non-singular,$(A, C) \in R$. Thus,$R$ is transitive.Since the relation is reflexive,symmetric,and transitive,it is an equivalence relation.

In the set of all $3 \times 3$ real matrices,a relation is defined as follows: $A$ matrix $A$ is related to a matrix $B$ if and only if there exists a non-singular $3 \times 3$ matrix $P$ such that $B = P^{-1} A P$. This relation is

Similar Questions

Let $R_{1} = \{(a, b) \in N \times N : |a - b| \leq 13\}$ and $R_{2} = \{(a, b) \in N \times N : |a - b| \neq 13\}$. Then on $N$:

In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation,it is sufficient,if $R$

Let $A$ be the set of all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ and $R$ be a relation on $A$ such that $R =\{( f , g ): f(0)= g (1) \text{ and } f(1)= g (0)\}$. Then $R$ is:

Let $A = \{1, 2, 3, 4\}$ and $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is:

Let $N$ be the set of natural numbers greater than $100$. Define the relation $R$ on $N$ by: $R = \{(x, y) \in N \times N : \text{the numbers } x \text{ and } y \text{ have at least two common divisors}\}.$ Then $R$ is-

Vedclass Test Series

Exam Paper Generator

Online Exam Module