Your thoughts on a proof of Matrix Operations

[Divergent13]

Greetings!

I am asked to do the following:

Simplify $$(A^{-1}B)^{-1}(C^{-1}A)^{-1}(B^{-1}C)^{-1}$$ for (n x n) invertible matrices A B and C.

You see, I was able to show that the result of this is simply the identity matrix $$I_n$$ by selecting 3 (2x2) matrices A B and C that were invertible, and just punched out the entire operation with them and ended up with the identity matrix I2... but clearly for an exam that would take way too long! How can I go about doing this using matrix properties? I am not sure how certain things cancel to get the Identity matrix...

Thanks for your help!

 

[Fredrik]

You should first prove that $$(XY)^{-1}=Y^{-1}X^{-1}$$, for any two non-singular ("invertible") n by n matrices X and Y. This is very easy, and when you've done it, the rest of your problem is also very easy.

 

[Divergent13]

So I understand that definition, and I obtain:

$$(B^{-1}A)(A^{-1}C)(C^{-1}B)$$

So I know that B^-1*B will yield the identity matrix, and the same identity matrices multipled by each other will be the same thing--- but in matrix mutliplication order is important--- so from here is it valid just to state this?

 

[Divergent13]

Would that qualify? I don't know if there's any "distributive" property i can use here.

 

[Hurkyl]

What about associative?

 

[Divergent13]

Got It Thank You!