Question

If two matrices \(A\) and \(B\) are invertible, \((AB)^{-1}=B^{-1}A^{-1}\). Is it true that if \(AB\) is invertible, then \(A\) and \(B\) are invertible?

Answer 1

i would say yes, under normal circumstances, that being the matrices are square over the reals

Answer 2

To me, it is true
Proof:
AB is invertible --> AB (AB)^- =I
AB B^-A^- =I iff BB^- =I and AA^-=I--> A, B are invertible.

Answer 3

not sure about that proof but since AB is invertible that means det(AB) is not zero
since the determinate of the product is the product of the determinates then they are also both non zero

Answer 4

Isn't that a proof?

Answer 5

:) when I took the course, I had the problem " Prove AB is invertible, then A, B are invertible"
Not exactly your question. hihihi

Answer 6

looks good to me

Answer 7

This is the question I am asking. I am asking "Given that AB is invertible, proof that A and B are both invertible."

Answer 8

So AB is invertible iff A and B are both invertible?

Answer 9

hahahaha.. if so, we may have the same professor.
And the answer is yes,

Answer 10

Does equivalence work?

Answer 11

you said AB invertible iff A and B are both invertible

Answer 12

A and B are invertible doesn't imply AB is invertible is it?