matrix help?

Question

A and B are nonsingular, suppose C=AB prove
C^ -1=B^-1A^-1
thanks in advance plz shows some works

Answer 1

you replace C=AB into the second equation
you have (AB)^-1=B^-1A^-1
right now you just prove RHS=LHS
multiply both sides by AB
so AB(AB)^-1=AB(B^-1A^-1)
I = AB(B^-1A^-1) cuz AB(AB)^-1= I bydefinition of inverse
so you prove RHS= I
first RHS=AB(B^-1A^-1)= A(BB^-1A^-1) cuz commute property
= AIA^-1= AA^-1= I by definition of inverse and A, B are nonsingular therefore A , B inverse is exist.
therefore I=I QED
2nd you need to prove
I=(B^-1A^-1)AB
that is processing the same.
i let you do the rest. if you stuck comeback . good luck

Answer 2

if it is true then

C^-1 C=B^-1 A^-1 A B

since for any non singular matrix M, M^-1 M=I

this reduces to I=I, hence the C^-1 you give is an inverse of C