Your thoughts on a proof of Matrix Operations

AI Thread Summary
The discussion focuses on simplifying the expression (A^{-1}B)^{-1}(C^{-1}A)^{-1}(B^{-1}C)^{-1} for invertible matrices A, B, and C. The user successfully demonstrated the result is the identity matrix I_n using specific 2x2 matrices but seeks a more efficient method using matrix properties. They are advised to first prove that (XY)^{-1}=Y^{-1}X^{-1} for invertible matrices, which simplifies the problem significantly. The conversation emphasizes the importance of understanding matrix multiplication properties, such as the order of multiplication and the use of the identity matrix. Ultimately, the user is encouraged to leverage these properties to streamline their proof for an exam setting.
Divergent13
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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!
 
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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.
 
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?
 
Would that qualify? I don't know if there's any "distributive" property i can use here.
 
What about associative?
 
Got It Thank You!
 
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