Singularity/Invertibility of Matrix Product

[Bipolarity]

Homework Statement

Suppose that A and B are square matrices of the same order. Prove that if AB is invertible, then A and B are both invertible.

Homework Equations

The Attempt at a Solution

I attempted to prove the contrapositive, i.e. if at least one of A,B is singular, then AB is singular. I proved that if B is singular, then AB is singular, but I have not been able to prove that if A is singular, then AB is singular.

I know that this somehow involves the notion that singular matrices such as C have non-trivial solutions to the homogenous system Cx = 0. But I can't apply it correctly because I don't know the direction to follow.

Any hints?

BiP

 

[Dick]

Homework Statement

Suppose that A and B are square matrices of the same order. Prove that if AB is invertible, then A and B are both invertible.

Homework Equations

The Attempt at a Solution

I attempted to prove the contrapositive, i.e. if at least one of A,B is singular, then AB is singular. I proved that if B is singular, then AB is singular, but I have not been able to prove that if A is singular, then AB is singular.

I know that this somehow involves the notion that singular matrices such as C have non-trivial solutions to the homogenous system Cx = 0. But I can't apply it correctly because I don't know the direction to follow.

Any hints?

BiP

If A is singular then Av=0 for some nonzero vector v. Since you've already shown B is nonsingular then v=Bu for some nonzero vector u, yes?