Answer
See below
Work Step by Step
If the inverse of $A^2$ is the matrix $B$ then $A^2B=I_n=BA^2$
Obtain: $A^{10}B^5=A^8(A^2B)B^4=A^8I_nB^4=A^8B^4=A^6(A^2B)B^3=A^6I_nB^3=A^6B^3=A^4(A^2B)B^2=A^4I_nB^2=A^4B^2=A^2(A^2B)B=A^2I_nB=A^2B=I_n$
and $B^5A^{10}=B^4(BA^2)A^8=B^4I_nA^8=B^4A^8=B^3(BA^2)A^6=B^3I_nA^6=B^3A^6=B^2(BA^2)A^4=B^2I_nA^4=B^2A^4=B(BA^2)A^2=BI_nA^2=BA^2=I_n$
The inverse ò the matrix $A^{10}$ is $B^5$
