Singularity/Invertibility of Matrix Product

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If the product of two square matrices A and B is invertible, both matrices must also be invertible. The discussion centers on proving the contrapositive: if either A or B is singular, then the product AB is singular. One participant successfully demonstrated that if B is singular, then AB is also singular but struggled to show the same for A. The conversation highlights the relationship between singular matrices and non-trivial solutions to the homogeneous equation. The proof requires a deeper understanding of matrix properties and their implications on invertibility.
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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
 
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Bipolarity said:

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?
 

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