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    • Thread Starter

    Hey guys, quick question about solving a matrix question:

    Suppose the matrix:

    \begin{bmatrix} cos w & p \\sin w & q \end{bmatrix}

    satisfies  A.A^t = I. Find all values of p and q.

    so \begin{bmatrix} cos w & p \\sin w & q \end{bmatrix} multiplied by \begin{bmatrix} cos w & sin w \\p & q \end{bmatrix} = I = \begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}

    therefore  cos^2w +p^2 = 1
     sin^2w + q^2 = 1
     sinw.cosw +pq = 0

    How do i then find all the solutions?? Thank you! +rep for helpful answers
    • PS Helper

    PS Helper
    Adding the first two equations gives \cos^2 w + \sin^2 w + p^2 + q^2 = 2, and hence using the identity \cos^2 x + \sin^2 x \equiv 1 we can simplify this to p^2 + q^2 = 1. Now, because this is the case, we can let p = \pm \sin \theta and q = \pm \cos \theta for some \theta that you have yet to discover (and you need to work out what the \pm signs are). But you can use the third equation together with the identity 2\sin x \cos x \equiv \sin 2x to find possible values of \theta in terms of w for each given choice of \pm. [Note that some of these might be equivalent; there are in fact only two cases you need to consider, not 4.]
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Updated: April 4, 2011

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