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Math Help - Matrix problem

  1. #1
    Junior Member rednest's Avatar
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    Post Matrix problem

    M =  \left[  \begin{array}{ c c }     4 & -5 \\     6 & -9  \end{array} \right]

    A transformation T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T.

    Find a cartesian equation of this line.
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  2. #2
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    Quote Originally Posted by rednest View Post
    M =  \left[  \begin{array}{ c c }     4 & -5 \\     6 & -9  \end{array} \right]

    A transformation T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T.

    Find a cartesian equation of this line.
    You need to find the eigenvalues of the matrix. so the two solutions to \left|  \begin{array}{ c c }     4 - \lambda & -5 \\     6 & -9 - \lambda \end{array} \right| =0 and one of the eigenvectors is mapped onto it self. the rest of the problem is fairly routine.

    Bobak
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  3. #3
    Moo
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    Hello,

    Quote Originally Posted by rednest View Post
    M =  \left[  \begin{array}{ c c }     4 & -5 \\     6 & -9  \end{array} \right]

    A transformation T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T.

    Find a cartesian equation of this line.
    Actually, you just have to find X=\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} which is the eigenvector associated to the eigenvalue 1, and such that :

    MX=X (translation from the text)

    and that's a characteristic of the eigenvalue 1 (when checking with bobak's formula, the determinant is indeed 0 when \lambda=1)
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