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A given force acts on two objects of different masses, one heavy and the other light , for the same duration of time. Which of the following statements is correct? [ and are the change in momenta of heavy and light body respectively]

A

B

C

D

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The same force acts at same time
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(Q1, Factor = .25) Let B be any invertible matrix. a) Prove that λ is an eigenpair of B if and only if 1/λ is an eigenpair of B^-1. Check examples first if you feel confused. b) Now, let A be an invertible matrix. Use the claim in (a) in order to find a formula for |A^-1| in terms of singular values of A. Explain. Hint: We assigned different notations to the matrix that appears in (a) and the matrix that appears in (b) for a reason! (Q2, Factor = .75) Let A be a symmetric matrix with the following special property: X ∈ Eo(A) - XA Since A ⊆ R, the above simply says that A does not have two different eigenvalues whose square is the same. a) For a 6x6 matrix A of your choice, use Matlab in order to find the spectrum and eigenvectors of A and the spectrum and eigenvectors of A'A = A^2 (your matrix A must be non-singular and cannot have any zero entry, and make sure that it satisfies the special condition above; any generic symmetric matrix will satisfy that condition). Based on that example, conjecture a general connection between the spectrum and the eigenvectors of a symmetric A that satisfies the above special condition and the spectrum and eigenvectors of A'A. The conjecture should be of the form (λ, x) is an eigenpair of A'A if and only if (λ, x) is an eigenpair of A. If you do not find any reasonable conjecture to make, run more examples. However, turn in the Matlab output of one of your tests only. b) Prove your conjecture from (a). Note that there are two parts in the proof (the "if" part and the "only if" part). c) In view of the above, state a theorem that connects the eigenvalues of a symmetric A to its singular values. Check your theorem against the matrix: [7 2] [7 2] d) Provide an example that shows that your theorem in (a) does not hold, in general, for symmetric matrices that do not satisfy the special condition above (Hint: you may find such example even in 2D, and the matrix may be chosen to be sensationally simple!)
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Question Text
A given force acts on two objects of different masses, one heavy and the other light , for the same duration of time. Which of the following statements is correct? [ and are the change in momenta of heavy and light body respectively]
TopicLaws of motion
SubjectPhysics
ClassClass 11
Answer TypeText solution:1
Upvotes55