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Homework Help: Inverse of matrix

  1. Sep 20, 2005 #1
    I'm having trouble with the following question - parts b, c and d. Can someone please help me out?

    Q. 3-d rotations - Consider the matrix:

    [tex]
    A\left( \theta \right) = \left[ {\begin{array}{*{20}c}
    {\cos \theta } & { - \sin \theta } & 0 \\
    {\sin \theta } & {\cos \theta } & 0 \\
    0 & 0 & 1 \\
    \end{array}} \right]
    [/tex]

    a) Evaluate det(A(theta)).
    b) Interpret geometrically the effect of multiplying a vector by A(theta).
    c) Show that [tex]A\left( \theta \right)A\left( \phi \right) = A\left( {\theta + \phi } \right)[/tex] and interpret this result.
    d) Use the previous part to find the inverse of A(theta). How does this compare with the transpose A(theta)^T - the transpose of A(theta).

    The answer to part 'a' is 1 which wasn't all that difficult to get. I'm not sure about part b. Ignoring the 3 row and 3rd column I have a 2 by 2 matrix which represents a rotation through an angle of theta in the anti-clockwise direction (in the x-y plane) but I'm not sure how to ineterpret the given 3 by 3 matrix.

    I could show the result of part 'c' but again, I don't know how to interpret the result. I think that it might just be the application of A(phi) followed by A(theta).

    I have no idea as to how to part 'd.'

    Any help would be good thanks.
     
  2. jcsd
  3. Sep 20, 2005 #2

    Galileo

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    Science Advisor
    Homework Helper

    You already identified it as a rotation if you consider the xy-plane. So you know what it does to the unit vectors x and y. What does the transformation do to the unit z vector?
    If you know the interpretation, the rest is easy.
     
  4. Sep 20, 2005 #3

    HallsofIvy

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    Science Advisor

    If multiplying a vector by a matrix causes a movement (as rotation), then the inverse matrix causes the opposite movement.
     
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