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Matrix/Diagnolisation watch

1. Sorry if it difficult to read, A^n is a 2x2 matrix. And the dots... signify a space to make it easier to read.

So we are given that :

A^n=PD^nP^-1, that

A^n=
[ 3(8^n) - 2(7^n) .... -8^n+7^n
6(8^n)-6(7^n) .... -2(8^n)+3(7^n) ]

Question: Find a matrix B such that B^2=A. [Hint:Guess a solution using the above]

I don't really understand when it means guessing a solution.
2. Hard not to simply give the answer.

Think of a value of n s.t. (A^n)^2 = A.

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