Answer
See answer below
Work Step by Step
Since the inverse of the matrix $A^3$ is $B^{-1}$, we can write the matrix $A^{15}$ as:
$A^{9}=(A^3)^3$
The inverse of $(A^9)^{-1}=((A^3)^3)^{-1}=((A^3)^{-1})^3=(B^{-1})^3=B^{-3}$
To check if the inverse of $A^9$ is $B^{-3}$:
so $A^9B^{-3}$
$=A^6A^3B^{-1}B^{-2}$
$=A^{6}(A^3B^{-1})B^{-2}$
$=A^6I_nB^{-2}$
$=A^{6}B^{-2}$
$=A^3(A^3B^{-1})B^{-1}$
$=A^3I_nB^{-1}$
$=A^3B^{-1}$
$=I_n$
Hence, $A^9$ is invertible with $B^{-3}$
