If two matrices A, B verify the relationship AB = BA

Prove that:

$$AB^{n} = B^{n}A$$ for any integer n.

Question

If two matrices A, B verify the relationship AB = BA

Prove that:

$$AB^{n} = B^{n}A$$     for any integer n.

jango_in_dtown · Answer

The proposition is ture for n=1, by the given condition. Let the proposition be true for some integer k>1. Then AB^k=B^kA
Now AB^(k+1)=AB^kB=(AB^k)B=B^kAB=B^kBA=B^(k+1)A/ Hence the proposition is true for k+1 whenever it is true for k. Hence by the method of induction....

trojanpoem · Answer

Thanks, nevertheless, My professor said : "Using induction is forbidden.",

jango_in_dtown · Answer

So let me rethink

jango_in_dtown · Answer

AB^n=AB...B=(BA)B...B=B(AB)B...B=B(BA)B...B=BB(AB)B...B=BB(BA)B...B=...=BB...B(AB)=BB...B(BA)=B^nA

trojanpoem · Answer

What did you do ? the so many B are confusing.

jango_in_dtown · Answer

just used the property BB...B (n times)=B^n

jango_in_dtown · Answer

and the given property AB=BA multiples times

trojanpoem · Answer

Nice.