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Test4 - TEST 4(Math 250 A B(Take Home 1 The Caley-Hamilton...

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Unformatted text preview: TEST 4 (Math 250 A, B) (Take Home) 1. The Caley-Hamilton Theorem provides a method for computing powers of a matrix A : (20 pts) e.g. Let A be a 2x2 matrix with characteristic equation 2 b ax x . Then, by the Caley-Hamilton Theorem, we have: 2 bI aA A which gives: bI aA A 2 . In other words, we just computed 2 A . Multiplying both sides by A , we get: bA aA A 2 3 So, we just computed 3 A , because the last equation is given in terms of the known quantities 2 A and A . Continuing in this way, we can compute n A , for some n , simply by expressing n A in terms of lower powers of A . Use the above procedure to compute 5 A for 2 1 6 3 A . 2. Consider the following Definition and the Theorem after that: (20 pts) Def .: Let A be a n x n matrix and an eigenvalue of A . The geometric multiplicity of is the dimension of the eigenspace E . The algebraic multiplicity of is the multiplicity of the root...
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Test4 - TEST 4(Math 250 A B(Take Home 1 The Caley-Hamilton...

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