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Matrix Series : Trace Log and Log Det

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Please see the attached file for the fully formatted problems.

Let K be an nxn matrix and  a small number. Imitating
.....
valid for small x, it is natural to define
....
Explain why this makes sense. Prove
trace log = log det(I + K)
Still with  small so that everything makes sense.
Hint: What is I − K +2K2 − 3K3 + · · · ?
Since this is an analysis problem, please be sure to be rigorous, and include as much detail as possible so that I can understand. Please also state if you are making use of some fact or theorem. Thanks!

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A matrix series problem is solved.

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