Question

$A$ and $B$ are two square matrices such that $A^{2}B=BA$ and if $(AB{)}^{10}=A^k{B}^{10}$, then $k$ is

• A. 1001
• B. 1023
• C. 1042
• D. None of these

Answer 1

Answer: (B)

$(AB)\cdot (AB)=A(BA)B=A^3{B}^2$
$(AB)(AB)(AB)=A^7{B}^3$
so $(AB{)}^n=A^{2^n-1}{B}^n$
so $k=2^{10}-1=1023$