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Math Help - Determine when vector is not in range of Matrix Transformation

  1. #1
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    Determine when vector is not in range of Matrix Transformation

    Let R^2->R^2 be the matrix transformation defined by f(x)=Ax, where A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}

    Determine k so that w=\begin{pmatrix}1\\1\\1\end{pmatrix} is not in the range of f.
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  2. #2
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    Quote Originally Posted by jennifer1004 View Post
    Let R^2->R^2 be the matrix transformation defined by f(x)=Ax, where A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}

    Determine k so that w=\begin{pmatrix}1\\1\\1\end{pmatrix} is not in the range of f.

    We want to determine k so that the column vector  w=\begin{pmatrix}1\\1\\1\end{pmatrix} is NOT in the range of f.

    So, we want k such that for any vector x,  Ax = \begin{pmatrix}1\\1\\1\end{pmatrix} NEVER HOLDS.

    That is,  \begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix} x = \begin{pmatrix}1\\1\\1\end{pmatrix} NEVER HOLDS.

    How can we assure this? What must k be/not be?
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