Originally Posted by

**HallsofIvy** I don't understand why you can do one and not the other (which is not really "more complicated").

First, do you understand that matrix multiplication is NOT commutative? That is, you need to be careful about on which side you multiply.

You have $\displaystyle A^{-1}BCD^{-1}A= I$

Okay, just as you did before, eliminate the $\displaystyle A^{-1}$ on the left of the left side by multiplying both sides by A **on the left**

$\displaystyle AA^{-1}BCD^{-1}A= AI$

$\displaystyle BCD^{-1}A= A$

Now eliminate the "A" on the right of the left side by multiplying both sides by $\displaystyle A^{-1}$ on the right

$\displaystyle BCD^{-1}AA^{-1}= AA^{-1}$

$\displaystyle BCD^{-1}= I$

Can you continue from here?