Let A and B be two non singular matrices of same order such that (A B)k=AkBk for consecutive positive integral values of k , then AB2A- 1 is equal to

Question

Let A and B be two non singular matrices of same order such that (A B)k=AkBk for consecutive positive integral values of k , then AB2A- 1 is equal to (A) A2 (B) B (C) A (D) B2

Tardigrade Admin · Accepted Answer

|A|≠ 0,|B|≠ 0 (given) (A B)3=A3B3 ldots (i) Also, (A B)3=(A B)2(A B)=A2B2AB ldots (i i) From (i) and (i i) AB2=B2A AB2A- 1=B2AA- 1=B2

Q. Let $A$ and $B$ be two non singular matrices of same order such that $(A B)^{k} = A^{k} B^{k}$ for consecutive positive integral values of $k$ , then $A B^{2} A^{- 1}$ is equal to

$A^{2}$

$B$

$A$

$B^{2}$

$∣ A ∣ \neq = 0, ∣ B ∣ \neq = 0$ (given)
$(A B)^{3} = A^{3} B^{3} \dots (i)$
Also, $(A B)^{3} = (A B)^{2} (A B) = A^{2} B^{2} A B \dots (ii)$
From $(i)$ and $(ii)$
$A B^{2} = B^{2} A$
$A B^{2} A^{- 1} = B^{2} A A^{- 1} = B^{2}$

Q. Let A and B be two non singular matrices of same order such that (AB)k=AkBk for consecutive positive integral values of k , then AB2A−1 is equal to

A2

B

A

B2

∣A∣=0,∣B∣=0 (given) (AB)3=A3B3…(i) Also, (AB)3=(AB)2(AB)=A2B2AB…(ii) From (i) and (ii) AB2=B2A AB2A−1=B2AA−1=B2

Q. Let $A$ and $B$ be two non singular matrices of same order such that $(A B)^{k} = A^{k} B^{k}$ for consecutive positive integral values of $k$ , then $A B^{2} A^{- 1}$ is equal to

$A^{2}$

$B$

$A$

$B^{2}$

$∣ A ∣ \neq = 0, ∣ B ∣ \neq = 0$ (given)
$(A B)^{3} = A^{3} B^{3} \dots (i)$
Also, $(A B)^{3} = (A B)^{2} (A B) = A^{2} B^{2} A B \dots (ii)$
From $(i)$ and $(ii)$
$A B^{2} = B^{2} A$
$A B^{2} A^{- 1} = B^{2} A A^{- 1} = B^{2}$