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set_4_problem

# set_4_problem - -2-3-1 1 2 1 3 3 2 satisﬁes the...

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Problem Set # 4 MMAE: 501 Engineering Analysis I Kevin W. Cassel Mechanical, Materials and Aerospace Engineering Department Illinois Institute of Technology 10 West 32nd Street Chicago, IL 60616 [email protected] Problem Reference Problem # 1: Hilderbrand, Chapter 1, Problem 85 Problem # 2: Hilderbrand, Chapter 1, Problem 88 Problem # 3: Jeffrey, Section 4.2, Problem 23 Problem # 4: Jeffrey, Section 4.2, Problem 26 Problem # 5: Jeffrey, Section 6.11, Problem 7 Problem # 6: Jeffrey, Section 6.11, Problem 13 c 2007 Kevin W. Cassel 1

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Problem # 1 Let A = " 2 1 1 2 # and B = A 5 - 3 A 4 + 2 A - I . (a) Determine the eigenvalues and corresponding eigenvectors of B . (b) Determine whether B is positive definite. Problem # 2 Solve the system dx 1 dt = x 1 + 2 x 2 dx 2 dt = 2 x 1 + x 2 subject to the initial conditions { x 1 (0) , x 2 (0) } = { c 1 , c 2 } . Problem # 3 Verify by direct calculation that the matrix A =
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Unformatted text preview: -2-3-1 1 2 1 3 3 2 satisﬁes the Cayley–Hamilton theorem. Problem # 4 For the matrix A = " 5 1 3-2 # determine its inverse using A-1 =-1 c n ( A n-1 + c 1 A n-2 + ··· + c n-1 I ) , recalling that the characteristic equation may be written in the form P ( λ ) = (-1) n [ λ n + c 1 λ n-1 + ... + c n-1 λ + c n ] = 0 . 2 Check the result by showing that AA-1 = I . Problem # 5 Find the general solution of the following system of equations by diagonalization x 1 =-10 x 1-18 x 2 + t x 2 = 6 x 1 + 11 x 2 + 3 . Problem # 6 Find the general solution of the following system of equations by diagonalization x 1 =-2 x 1 + 2 x 2 + 2 x 3 + sin t x 2 =-x 2 + 3 x 3 =-2 x 1 + 4 x 2 + 3 x 3 . 3...
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