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H51 - Math 330 Homework 5.1(Pages 308,309(2 Is = 2 an...

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Math 330 Homework 5.1 (Pages 308,309) (2) Is λ = 2 an eigenvalue of A = bracketleftBigg 3 2 2 8 bracketrightBigg ? Why or why not? SOLUTION: If there are nonzero vectors vectorx so that Avectorx = 2 vectorx then it means that A - 2 I has a nontrivial null space: A - 2 I = bracketleftBigg 1 2 2 6 bracketrightBigg However, this matrix is invertible since the rows are not multiples of each other. Therefore, 2 is not an eigenvalue. (4) Is bracketleftBigg - 1 + 2 1 bracketrightBigg an eigenvector of bracketleftBigg 2 1 1 4 bracketrightBigg ? If so, find the eigenvalue. SOLUTION: We check the matrix multiplication bracketleftBigg 2 1 1 4 bracketrightBigg bracketleftBigg - 1 + 2 1 bracketrightBigg = bracketleftBigg 2( - 1 + 2) + 1 ( - 1 + 2) + 4 bracketrightBigg = bracketleftBigg - 1 + 2 2 3 + 2 bracketrightBigg If this is equivalent to matrix multiplication by some scalar λ then the second coordinate tells us that we must have λ = 3 + 2. If this is indeed the case then the same multiplier must work for the first coordinate and so we check
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