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LINEAR ALGEBRA: A is a 4X4 matrix, b&c are in R^4, Ax=b has uniq. soln. - Does Ax=c?

  1. Sep 16, 2006 #1
    Let [itex]A[/itex] be a 4 X 4 matrix, and let [itex]\overrightarrow{b}[/itex] and [itex]\overrightarrow{c}[/itex] be two vectors in [itex]\mathbb{R}^4[/itex]. We are told that the system [itex]A\,\overrightarrow{x}\,=\,\overrightarrow{b}[/itex] has a unique solution. What can you say about the number of solutions of the system [itex]A\,\overrightarrow{x}\,=\,\overrightarrow{c}[/itex]?

    MY ANSWER:

    The only way that the system [itex]A\,\overrightarrow{x}\,=\,\overrightarrow{c}[/itex] has a unique solution is if A is invertable.

    A is to be invertable.

    [itex]A\,\overrightarrow{x}\,=\,\overrightarrow{c}[/itex] has a unique solution.

    But maybe the solution set [itex]\overrightarrow{c}[/itex] does not have a unique solution because A is not truely invertable?
     
    Last edited: Sep 17, 2006
  2. jcsd
  3. Sep 17, 2006 #2

    Galileo

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    If Ax=b has a unique solution, what does that tell you about the matrix A?
     
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