A and B are two n n matrices a Prove that if A and B are bo

A and B are two n × n matrices.

a. Prove that if A and B are both invertible, then AB is also invertible.

b. Prove ( without using determinants) that if AB is invertible then A and B are both invertible.

Solution

a)

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:

(AB)(B^-1A^-1)=ABB^-1A^-1=A(BB^-1)A^-1=AIA^-1=AA^-1=I.

We used the associativity of the product of matrices, the definition of an inverse and the fact that IA=AI=A for every matrix A.

b) If AB is invertible, then AB(AB)-1=I, which means that A has the inverse B(AB)-1. Similarly, (AB)-1AB=I means that B has the inverse (AB)-1A.
There are long methods as well to prove this. This is the shortest logical explanation.

anither method

You also can show that if C = AB is invertible, then A is right invertible, and B is left invertible. Since A and B are square matrices, we can conclude by counting pivots, that they are invertible.