Matrix

True.

By expanding the term on LHS,
we can have terms A^n +nC1 A^(n-1) B+...+b^n

For the terms in the middle, they must have AB or BA, which are equal to 0.
Therefore, all the terms are zero except the first and last terms.

The answer is then A^n+B^n

I agree with cat cat as the question requires you to prove it.
So, mathematical induction is more convincing.

True.

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圖片參考：https://s.yimg.com/rk/HA00430218/o/655125844.png


2014-07-03 11:24:59 補充：
No offence.

The proof is understandable but it is not rigorous in a mathematical sense.