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    • Thread Starter

    what does the det of a 2x2 matrix represent?

    theres a question :

    matrix A :

    1 -2
    2 1

    give a full geometrical description of T

    det of T is obviously 5.
    then all 4 numbers divide by root 5 to find a rotation matrix

    the answer said the enlargement sf is root 5.

    1.does this rule apply to all 2x2 matrix (root of det = enlargement sf)???
    2.is it area enlargement sf or just normal enlargement sf?

    thx a lot
    • Wiki Support Team

    Wiki Support Team
    The matrix A is \sqrt{5} \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} for some theta, i.e. a rotation followed by an enlargement by a length scale factor of sqrt(5). This certainly doesn't apply to all matrices (most matrices will scale in different directions differently, and they won't be obvious directions), but something similar does. Take the unit square (i.e. the one with corners at (0,0), (0,1), (1,0), (1,1)), and apply the matrix M to it; then the resulting area will be (det M).
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Updated: January 22, 2010

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