if det(A)=0 , then AB is singular

Prove the following:

1- if det(A)=0 , then AB is singular .

2- if A and B are invertible then AB is invertible . with (AB)^-1 = B^-1 A^-1

3- if A is invertible and B , C are inverses of A , the B=C .

4- if A is invertible then det (A^-1)= 1/det (A) .