For square matrices A & B that satisfy $AB + BA=0$, how to show that $A^2 B^3 = B^3 A^2$?

Question

How can we prove that $A^2 B^3 = B^3 A^2$?

Answer 1

$AB+BA=0$ means that $A$ and $B$ anti-commute: $AB=-BA$.

Next, we can prove that in this case $A^2$ and $B$ commute: $$A^2B=AAB=A(AB)=-A(BA)=-(AB)A=-(-BA)A=BAA=BA^2$$

(notice how the sign changed two times; the same way we can prove that $A^3$ and $B$ anti-commute, $A^4$ and $B$ commute, and so on)

$A^2B=BA^2$ means that we can freely permute $A^2$ with $B$. Now the result easily follows:

$$A^2B^3=A^2BBB=BA^2BB=BBA^2B=BBBA^2=B^3A^2$$

Answer 2

$$A^2 B^3 = AABB^2 = -ABAB^2 = BAAB^2 = -BABAB = B^2AAB = -B^2ABA = B^3A^2$$