If ABCD = I express B^(-1) in terms of A,C, and D (Linear algebra. Attempt inside)

I'm looking for help to this problem. Here is my attempt:

I being the identity matrix

A (B B^(-1)) C D = B^(-1) I
so A I C D = B^(-1)
so A C D = B^(-1)

Am I wrong? If so could you please point out where and how I can solve this problem. Thank you.

Yes, it is wrong. The error is in the first deduction: the equalty ABCD = I does not imply that A (BB^(-1)) CD = B^(-1) I.

I don't know how to "explain" this except to point out that there's no reason why you would expect that to be true. It may help to give examples of things that you *can* expect to be true. All of the ones I can think of are based on a simple principle: if two things are equal, then if you do the same thing to both of them, the resulting two things are equal. So if P and Q are matrices and P = Q, you can deduce that P + 5Q = Q + 5Q (for example: adding 5Q to equal things produces equal things), and that P - P = Q - P (subtracting P from equal things produces equal things), and that PM = QM for any matrix M (multiplying equal things on the right by a matrix M produces equal things), and that MP = MQ for any matrix M (multiplying equal things on the left by a matrix M produces equal things). Those are example deductions you can make from a matrix equality P = Q.

So if ABCD = I you can deduce, among other things, that ABCD - I = I - I, or that ABCD - ABCD = I - ABCD, or that AABCD = AI, and so on, because these are all instances of doing the same thing to equal things. As far as bringing B^(-1) into it goes, you could multiply both sides on the left by B^(-1), deducing B^(-1) ABCD = B^(-1) I, or multiply both sides on the right by B^(-1), deducing ABCDB^(-1) = I B^(-1). Both of these new equalities are true (whether or not they are useful for your purpose is a separate issue: but they are nevertheless true). So you can introduce B^(-1) on the left of both sides, or on the right of both sides. There's no general principle of matrix algebra that lets you put a B^(-1) in wherever you want into one side of a matrix equality as long as you put a B^(-1) somewhere else on the other side.