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Determinants, Cofactors and Permutations

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Suppose An is the n by n tridiagonal matrix with 1's everywhere on the three diagonals...

Let Dn be the determinant of An; we want to find it.
(a) Expand in cofactors along the first row of An to show that Dn = Dn-1 - Dn-2

(b) Starting from D1 = 1 and D2 = 0 find D3, D4, ..., D8. By noticing how these numbers cycle around (with what period?) find D1000.

Suppose the permutation S takes (1,2,3,4,5) to (5,4,3,2,1).
(a) What does S2 do to (1,2,3,4,5)?

(b) What does S-1 do to (1,2,3,4,5)?

Please see attached for full question.

Gilbert Strang's Linear Algebra and its Applications, 3rd edition.

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Question #1
a) It is easy to know that , . Now we consider . We know that for the matrix , , has the following form

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