Answer
See answer below
Work Step by Step
Since the inverse of the matrix $A^5$ is $B^3$, we can write the matrix $A^{15}$ as:
$A^{15}=(A^5)^3$
The inverse of $(A^{15})^{-1}=((A^5)^3)^{-1}=((A^5)^{-1})^3=(B^3)^3=B^9$
Check:
so $A^{15}B^9$
$=A^{10}A^5B^3B^6$
$=A^{10}(A^5B^3)B^6$
$=A^{10}I_nB^6$
$=A^{5}A^5B^3B^3$
$=A^5(A^5B^3)B^3$
$=A^5I_nB^3$
$=A^5B^3$
$=I_n$
Hence, $A^{15}$ is invertible with $B^9$
