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Thread: Matrix problem

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

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

    A transformation $\displaystyle T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ 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 $\displaystyle T$.

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

    A transformation $\displaystyle T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ 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 $\displaystyle T$.

    Find a cartesian equation of this line.
    You need to find the eigenvalues of the matrix. so the two solutions to $\displaystyle \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 = $\displaystyle \left[ \begin{array}{ c c } 4 & -5 \\ 6 & -9 \end{array} \right]$

    A transformation $\displaystyle T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ 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 $\displaystyle T$.

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

    $\displaystyle MX=X$ (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 $\displaystyle \lambda=1$)
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