082nd_tut10sol - MATH1111/2008-09/Tutorial X Solution 1...

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MATH1111/2008-09/Tutorial X Solution 1 Tutorial X Suggested Solution 1. Let A = 1 1 1 0 2 - 1 0 - 3 0 . (a) Find the characteristic polynomial, eigenvalues and eigenvectors for A . (b) Diagonalize A . (i.e. Find P and a diagonal matrix D such that P - 1 AP = D .) (c) Use Part (b) to evaluate A 4 . Ans . (a) The characteristic polynomial is p ( x ) = - ( x - 1)( x +1)( x - 3). The eigenvalues are - 1 , 1 , 3. The eigenvectors belonging to λ = - 1 is α ( - 2 1 3) T , where α 6 = 0. The eigenvectors belonging to λ = 1 is α (1 0 0) T , where α 6 = 0. The eigenvectors belonging to λ = 3 is α (0 1 - 1) T , where α 6 = 0. (b) By part (a), we set P = 1 - 2 0 0 1 1 0 3 - 1 . Then, P - 1 AP = 1 0 0 0 - 1 0 0 0 3 = D. (c) By part (b), we get ( P - 1 AP ) 4 = D 4 , and note that ( P - 1 AP ) 4 = P - 1 APP - 1 APP - 1 APP - 1 AP = P - 1 A 4 P and D 4 = 1 0 0 0 1 0 0 0 3 4 . Thus, A 4 = PD 4 P - 1 = 1 - 2 0 0 1 1 0 3 - 1 1 0 0 0 1 0 0 0 3 4 1 1 2 1 2 0 1 4 1 4 0 3 4 - 1 4 = 1 0 0 0 61 - 20 0 - 60 21 .
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MATH1111/2008-09/Tutorial X Solution 2 2. Under what conditions on a, b, c will the matrix B =
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