Let A and B be nn matricesIf AB is invertible then A and B n

Let A and B be n×n matrices.If AB is invertible, then A and B need not be invertible. True or False?Provide proof.

Ans. False,

Remember:

A square matrix A is invertible if and only if x = 0 is the only solution of the matrix equation Ax = 0

Now lets Assume that there are two matrices C and D.

C = B*(AB)^-1

D = (AB)^-1*A

Then AC = A*B*(AB)^-1 = (AB)*(AB)^-1

DB = (AB)^-1*A*B = (AB)^-1*(AB)

And we know that

(AB)*(AB)^-1 = I

This means that C = A^-1 and D = B^-1

So to exist A^-1 and B^-1, A and B need to be invertible.