A and B are two non-zero square matrices such that ${A^2}B=BA$ and if $(AB)^{10} = {A^k}{B^{10}}$ then the value of k is?
Attempt:
Tried solving for lower powers of $AB$ and observing a pattern
$AB=AB$
$(AB)^{2} = {A^3}{B^{2}}$
$(AB)^{3} = {A^7}{B^{3}}$
$(AB)^{4} = {A^{15}}{B^{4}}$
The series follows a pattern $(AB)^{n} = {A^{2^n-1}}{B^{n}}$.
Hence $(AB)^{10} = {A^{1023}}{B^{10}}$
Is there any general method to obtain the $n^{th}$ term without inspection of initial values?
Now that you know what the answer is, try induction. $$(AB)^{n+1}=(AB)^n(AB)=A^xB^nAB$$ So you first need another induction to find $$B^nA=A^yB^n$$