Consider the following relation R on the set of real square matrices of order 3.
R = {(A, B)|A = P –1 BP for some invertible matrix P}.
Statement -1 : R is equivalence relation.
Statement - 2 : For any two invertible 3 × 3 matrices M and N, (MN) –1 = N –1 M –1 .
Text Solution
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for reflexive
(A , A) ∈ R ⇒ A = P –1 A P
which is true for P = I
∴ reflexive
for symmetry
As (A, B) ∈ R for matrix P
A = P –1 BP ⇒ PA = PP –1 BP ⇒ PAP –1 = IBPP –1
⇒ PAP –1 = IBI ⇒ PAP –1 = B ⇒ B = PAP –1
∴ (B, A) ∈ R for matrix P –1 ∴ R is symmetric
for transitivity
A = P –1 BP and B = P –1 CP ⇒ A = P –1 (P –1 CP)P
⇒ A = (P –1 ) 2 CP 2 ⇒ A = (P 2 ) –1 C(P 2 )
∴ (A, C) ∈ R for matrix P 2 ∴ R is transitive
so R is equivalence
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