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## Homework Statement

Let A, B, C, D be

*nxn*complex matrices such that AB and CD are Hermitian, i.e., (AB)*=AB and (CD)*=CD.

Show that AD-B*C*=I implies that DA-BC=I

The symbol * indicates the conjugate transpose of a matrix, i.e., M* is the conjugate transpose of M.

I refers to the identity.

## Homework Equations

Wikipedia says that if A and B are Hermitian, AB is only Hermitian if AB=BA. I'm not sure if this is supposed to be a biconditional, and AB=BA indicates that AB is Hermitian.

I don't know what else may be relevant to proving this.

## The Attempt at a Solution

I'm not sure where to start. Just because AB and CD are Hermitian, do we know that A, B, C, and D individually are hermitian? I suppose that if we knew that, and we knew (somehow) that AD=DA, the proof is trivial. But alas, we don't know those things, do we?

Thank you for your help. I appreciate it.

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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