- #1
Saladsamurai
- 3,020
- 7
Show if A,B,C are invertible matrices of same size...
I know, I know. I should be awesome at these by now...
Show if A,B,C are invertible matrices of the same size, than [itex](ABC)^{-1}=C^{-1}B^{-1}A^{-1}[/itex]
If:
[itex]AA^{-1}=I[/itex]
[itex]BB^{-1}=I[/itex]
[itex]CC^{-1}=I[/itex]
I am not sure where to go from here. I don't think I have any more definitions or product rules to incorporate.
It almost seems as if I would FIRST have to show that (ABC) is invertible to begin with. Then I can use the fact that (ABC)(ABC)^{-1}=I to discover how (ABC)^{-1} MUST be arranged in order for the product of the two to yield I.
Does that sound like a good place to start? Proving if A,B, and C are invertible, then (ABC) is too?
I know, I know. I should be awesome at these by now...
Homework Statement
Show if A,B,C are invertible matrices of the same size, than [itex](ABC)^{-1}=C^{-1}B^{-1}A^{-1}[/itex]
The Attempt at a Solution
Given some matrix A,[itex]AA^{-1}=I[/itex]If:
[itex]AA^{-1}=I[/itex]
[itex]BB^{-1}=I[/itex]
[itex]CC^{-1}=I[/itex]
I am not sure where to go from here. I don't think I have any more definitions or product rules to incorporate.
It almost seems as if I would FIRST have to show that (ABC) is invertible to begin with. Then I can use the fact that (ABC)(ABC)^{-1}=I to discover how (ABC)^{-1} MUST be arranged in order for the product of the two to yield I.
Does that sound like a good place to start? Proving if A,B, and C are invertible, then (ABC) is too?