Define a sequence of real numbers a_0, a_1, a_2, ... by a_0 = 0, a_1 = 0, a_2 = 1, and a_(n+3) = -a_ ...

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Define a sequence of real numbers a_0, a_1, a_2, ... by a_0 = 0, a_1 = 0, a_2 = 1, and a_(n+3) = -a_(n+2) + 4 a_(n+1) + 4 a_n for all n. (a) Write down a 3x3 matrix A such that the column vector (a_(n+1), a_(n+2), a_(n+3)) equals A times the column vector (a_n, a_(n+1), a_(n+2)) for all n. (b) Without computing any eigenvectors, explain why A is diagonalizable. (c) Diagonalize A. (d) Use the previous parts to derive a non-recursive formula for a_n in terms of n. (e) Is A orthogonally diagonalizable?