Question

6.10
EXAMPLE Consider the following relation on the set of real
square matrices of order 3.
9R={(A,B):A-P-BP for some mortible matrix P1
Statement-1: R is an equilar relation
Statement-2: For any two invertible 3 x 3 matrices M and N.
(MN) NM
(a) 1 (6) 2 () 3 (d) 4​

Answer 1

Answer:

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Answer 2

Answer:

Since it is not specified what is 1,2,3 and 4 match the answer with the option given.

Both the statements are corrent but statement 2 is not the correct explanation of statement 1.

Explanation:

R is an equivlence relation.

Reflexive:

R is a reflexive relation for P = I.

A=$P^{-1}AP$ for P=I

Symmetric:

Given A =$P^{-1}BP$

This emplies: B = PA$P^{-1}$

let there be another matrix Q, such that Q = $P^{-1}$

then the eqation becomes: B = $Q^{-1}AQ$

Thus R is symmetric

Transitive:

Given A =$P^{-1}BP$

let B = $M^{-1}CM$

substituting the value of B from second equn in the first eqation

A = $P^{-1}$$(M^{-1}CM)$$P$

A= $(P^{-1}M^{-1})C(MP)$

A =$(MP)^{-1}$$C(MP)$

Let there be a matrix N= MP

Then

A = $N^{-1}CN$

This shows that R is transitive aswell.

Since R is transitive , symmetric and reflexive we can call it a Equavalence realtion.

Statement 2:

Statement 2 is correct but it has nothing to do with Statement 1, so it is not the correct explanation of the first statement.

Hope this helps!!!