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Assignment 7 solution

# Assignment 7 solution - MATH 223 Linear Algebra Winter 2008...

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MATH 223, Linear Algebra Winter, 2008 Solutions to Assignment 7 1. The Cayley-Hamilton Theorem says that if A is a square matrix and χ A ( x ) is the characteristic polynomial of A , then χ A ( A ) = 0. I will not be proving this in general, because we don’t really need it, but the case when A is 2 × 2 is straightforward enough. Prove it for 2 × 2 matrices. Solution: If A = a b c d , then χ A ( x ) = x 2 - ( a + d ) x + ( ad - bd ). So χ A ( A ) = A 2 - ( a + d ) A + ( ad - bc ) I = a 2 + bc ab + bd ac + cd bc + d 2 - a 2 + ad ab + bd ac + cd ad + d 2 + ad - bc 0 0 ad - bc = 0 . [This shows that the minimal polynomial of a 2 × 2 matrix has degree at most 2; in fact, the degree is exactly two unless the matrix is λI for some scalar λ , and so except in this case, χ A = min A for 2 × 2 matrices. It would have been useful to know this for assignment 6.] 2. When a sequence is defined in the style of the Fibonacci sequence, an explicit formula can be found using linear algebra methods. Suppose that a and b , c and d are scalars and we define x 0 = a , x 1 = b , and x n +2 = cx n + dx n +1 for n 0. (So x 2 = ac + bd , x 3 = bc + acd + bd 2 and so on.) (a) Show that, if we define the sequence x n (for n a natural number) as above, then for any n , x n +1

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