Answer
See below
Work Step by Step
If the inverse of $A^3$ is the matrix $B^2$ then $A^9=(A^3)^3$
Thus $(A^9)^{-1}=((A^3)^3)^{-1}=((A^3)^{-1})^3=(B^2)^3=B^6$
Obtain: $A^9B^6=A^6A^3B^2B^4=A^6(A^3B^2)B^4=A^6(I_n)B^4=A^6B^4=A^3(A^3B^2)B^2=A^3(I_n)B^2=A^3B^2=I_n$
and $B^6A^9=B^4B^2A^3A^6=B^4(B^2A^3)A^6=B^4(I_n)A^6=B^4A^6=B^2(B^2A^3)A^3=B^2(I_n)A^3=B^2A^3=I_n$
The inverse ò the matrix $A^{9}$ is $B^6$
