Assignment 7

Assignment 7 - 2 . Show that there are scalars α and β...

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MATH 223, Linear Algebra Winter, 2008 Assignment 7, due in class Wednesday, March 12, 2008 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. 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 x n = ± d c 1 0 n ± b a . (b) Suppose that the matrix ± d c 1 0 has distinct eigenvalues λ 1 and λ
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Unformatted text preview: 2 . Show that there are scalars α and β such that x n = α ( λ 1 ) n + β ( λ 2 ) n for all n . (c) Give an explicit formula for each of the following recursively defined sequences. x = 2 , x 1 = 3 , x n +2 = x n +1 + 6 x n . x =-4 , x 1 = 2 , x n +2 = 8 x n +1-20 x n . 3. Calculate (by hand, SVP) det 7-1 3-8 5 13-4-6 1 3 2-2 4 4 1 5 . 4. Suppose that A = a b c d e f g h j and B = d * a g e * b h f * c j are matrices with complex entries, and det ( A ) = 4 i , det ( B ) =-2 + i . What is det ( C ) if C = g h j 2 id-d * 2 ie-e * 2 if-f * (1 + i ) a (1 + i ) b (1 + i ) c ? 1...
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This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.

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