Theorem. The following properties hold:
that is the inverse of the product is the product of inverses in the opposite order. In particular
1. Indeed if AB=I, CA=I then
3. We need to prove that if A and B are invertible square matrices then
B-1A-1 is the inverse of AB. Let us denote B-1A-1 by C (we always have to
denote the things we are working with). Then by definition of the inverse
we need to show that (AB)C=C(AB)=I. Substituting B-1A-1 for C we get:
We used the
associativity of the product of matrices, the definition of
and the fact that IA=AI=A for every matrix A.
Other properties were left as exercises.