(AB) (B-1 A-1) = A (BB-1) A-1 , by associativity.
= A I A-1 = AA-1 = I
The answer to the question shows that:
(AB)-1 = B-1 A-1
Notice that the order of the matrices has been reversed on the right of the "=" . Another sometimes useful property is:
(A-1)T = (AT)-1