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I am asked to do the following:

Simplify [tex](A^{-1}B)^{-1}(C^{-1}A)^{-1}(B^{-1}C)^{-1}[/tex] 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 [tex]I_n[/tex] 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!!