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Permutable matrices

  1. Oct 3, 2006 #1
    I have this question as homework from my Algebra class:
    A square matrix X is called exchangeable with A if AX=XA.Determine the set of permutable matrices with [​IMG].

    My question is,how do I find that set?I know that a matrix to be permutable all rows and columns must be the same and that a square matrix is composed by the same number of rows and columns.
    Thanks in advance for the help!
     
  2. jcsd
  3. Oct 4, 2006 #2

    HallsofIvy

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    ?? You defined "exchangeable" with A and then asked for "permutable" with A?? Then you defined "permutable" matrix without any reference to a matrix A?? What am I missing?


    If you want to find all matrices that are "exchangeable" with A (standard terminology: "that commute with A"), then look at
    [tex]\left[\begin{array}{cc}a & b \\ c & d \end{array}\right]\left[\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right]= \left[\begin{array}{cc}1 & 1 \\0 & 1\end{array}\right]\left[\begin{array}{cc}a & b \\c & d\end{array}\right][/tex]

    If I understand your definition of "permutable" correctly: "all rows and columns must be the same", then all 2 by 2 permutable matrices are of the form
    [tex]\left[\begin{array}{cc}a & a \\ a & a\end{array}\right][/tex]
    and the only "permutable" matrix that is "exchangeable" with A is
    [tex]\left[\begin{array}{cc}0 & 0 \\ 0 & 0 \end {array} \right][/tex]
     
  4. Oct 4, 2006 #3
    ...

    Sorry for the mistypeing!:redface: When I said "exchengeable I meant to say permutable,so it would be like:

    "A square matrix X is called permutable with A if AX=XA..."
     
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