Question

Suppose A and B be two non-singular matrices, such that $AB=B{A}^m$ and $B^n=I$ and $A^p=I$
If m = 3 and n = 4, then p is equal to

• A. 80
• B. 81
• C. 83
• D. 127

Answer 1

The correct option is A 80

$B=A^{-1}BAAA{B}^4=\left(A^{-1}BAAA\right)\left(A^{-1}BAAA\right)\left(A^{-1}BAAA\right)\left(A^{-1}BAAA\right)\Rightarrow I=A^{-1}BA\left(AB\right)AABAABAAA=A^{-1}BA(BA{)}^{3}AABAABAAA=A^{-1}BB{A}^{8}B{A}^{2}B{A}^{2}B{A}^3=A^{-1}{B}^{2}B{A}^{24}{A}^{2}B{A}^3$ $[A^{k}B=B{A}^{3k}]$
$=A^{-1}{B}^{3}B{A}^{78}{A}^3=A^{-1}{B}^{4}B{A}^{81}=A^{-1}I{A}^{81}=A^{80}$