Q. Let $A,B$ be two matrices such that they commute, then for any positive integer $n$, $AB^{n} = B^{n}\ A$ (ii) $(AB)^{n} = A^{n}B^{n}$
A) Only (i) is correct
B) Both (i) and (ii) are correct
C) Only (ii) is correct
D) None of (i) and (ii) is correct
Explanation
(b)
$AB^{n} = ABBBBB\cdots B$
$= (AB)BBB\cdots B$
$= B(AB)BBB\cdots B$
$= BB(AB)BB\cdots B$
$\vdots$
$= B^{n}A$
$(AB)^{n} = (AB)(AB)(AB)\cdots(AB)$
$= A(BA)(BA)(BA)\cdots(BA)B$
$= A(AB)(AB)(AB)\cdots(AB)B$
$= A^{2}(BA)(BA)(BA)\cdots(BA)B^{2}$
$= A^{3}(BA)(BA)(BA)\cdots(BA)B^{3}$
$\vdots$
$= A^{n}B^{n}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.