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Matrix inverse

  1. Mar 28, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that the matrix [cos(theta), -sin(theta); sin(theta), cos(theta)] is invertible, regardless of the value of theta

    2. Relevant equations

    Identity matrix, elementary row operations

    3. The attempt at a solution

    I have the basic idea as to how to go about this; (let the above matrix = A)

    - form an augmented matrix with the identity matrix, eg. [A|I]

    - perform row operations (forward and backwards elimination) until matrix looks like [I|A^-1]

    However, i'm at a loss as to how to perform these operations with the trigonometric values instead of numbers.

    Just a push in the right direction would be greatly appreciated, i'd like to solve this myself
     
  2. jcsd
  3. Mar 28, 2008 #2

    morphism

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    Try to think about this matrix geometrically. Can you guess what a possible inverse of it is? Verify that your guess is an actual inverse.
     
  4. Mar 28, 2008 #3

    HallsofIvy

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    The problem doesn't actually ask you to find the inverse- just to show that it exists. Are you aware that a matrix is invertible if and only if its determinant is not 0? What is the determinant of this matrix?

    Actually, it's not that hard to find the inverse they way you are doing it- just tedious. It turns out to be surprisingly easy and morphism's suggestion shows why.
     
  5. Mar 28, 2008 #4

    tiny-tim

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    … just a gentle push …

    Hi nk735! :smile:

    Try:
    (cosA sinA (cosB sinB
    -sinA cosA) x -sinB cosB).

    The (1,1) term will be cosAcosB + sinA(-sinB), = … ?

    So the whole matrix is … ? :smile:
     
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