Proving Invertibility of Matrices A & B: AB invertible

[mathboy]

Question: Let A and B be nxn matrices such that AB is invertible. Prove that A and B are invertible.

All I have so far is that there exists a matrix C such that
(AB)C = I and C(AB) = I.

How do I use this to show that there exists D such that AD = DA = I and that there exists E such that BE = EB = I ?

 

[mathboy]

If no such D and E exist, then there are no such D and E such that

I = EB = E(DA)B = (ED)(AB),

contradicting the existence of C. Is that correct?

 

[HallsofIvy]

Are you specifically looking for a "direct" proof? If not, then the simplest argument is: A matrix is invertible if and only if its determinant is non-zero. If AB is invertible, its determinant is non-zero. Also det(AB)= det(A)det(B). For a product of two numbers to be non-zero, neither can be zero- det(A) is non-zero so A is invertible; det(B) is non-zero so B is invertible.

I would agree that a direct proof, not using the determinant, would be preferable. For that, you will have to be careful. It is NOT true, in general, that if, for two functions f and g, f(g(x)) has an inverse, then f and g separately have inverses. To prove this for matrices (i.e. representing linear transformations) you will need to use the "linearity". In particular, if B is NOT invertible, then there must exist a non-zero vector, v, such that Bv= 0. But then ABv= A(Bv)= A0= 0.

 

[HallsofIvy]

If no such D and E exist, then there are no such D and E such that

I = EB = E(DA)B = (ED)(AB),

contradicting the existence of C. Is that correct?

I'm not sure it makes a lot of sense to say if "no such D and E" exist, and then write an equation with D and E! As I said before, the statement "if f(g(x)) has an inverse, then f(x) and g(x) must have inverses", for general functions, f and g, is NOT true. Let g:{a, b, c}-> {x} be defined by g(a)= x, g(b)= x, g(c)= x and f:{x}-> {y} be defined by f(x)= y. Then f(g) has no inverse because it maps all of {a, b, c} into y and is not "one to one". But g DOES have an inverse.

 

[Dmak]

Suppose B is singular then there exists a nonzero vector v such that Bv = 0 hence
(AB)v = A(Bv) = A(0) = 0 but AB is nonsingular so v must equal zero.

Similar situation for A as well.