Question

Let $A$ and $B$ are two non-singular matrices such that $AB=B{A}^2,B^4=I$ and $A^k=I,$ then $k$ can be equal to

• A. $5$
• B. $10$
• C. $15$
• D. $16$

Answer 1

Answer: (C)

$A{B}^4=AB(B)(B)(B)$
$=B{A}^2(B)(B)(B)$
$=BA(AB)BB$
$=BA(B{A}^2)BB$
$=B(AB)(A^{2}BB)$
$=BB{A}^{4}BB$
$=B^{2}B{A}^{8}B=B^4{A}^{16}$
$\Rightarrow A=A^{16}\Rightarrow A^{15}=I$