Suppose A and B are two square matrices of the same size Sho

Suppose A and B are two square matrices of the same size. Show that AB is invertible if and only if A and B are both invertible.

Solution

Then, since A is invertible, then there exists a matrix A^-1 such that AA^-1 = I and A^-1A = I, where I is the identity matrix.

Similarly, since B is invertible, then there exists a matrix B^-1 such that BB^-1 = I and B^-1B = I.

To show that AB is invertible, all that one has to do is to demonstrate that it has an inverse; that is, we must exhibit a matrix C such that (AB)C = I, and C(AB) = I.

Selecting B^-1A^-1 to be the matrix C works, because

(AB)(B^-1A^-1) = A(BB^-1)A^-1 = A(I)A^-1 = (AI)A^-1 = AA^-1 = I; and
(B^-1A^-1)AB = B^-1(A^-1A)B = B^-1(I)B = (B^-1I)B = B^-1B = I.

So AB is invertible and B^-1A^-1 is the inverse of AB; in other words, B^-1A^-1 = (AB)^-1.