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Math Help - Find all matrices given constraint.

  1. #1
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    Find all matrices given constraint.

    Let B = \begin{bmatrix}1&0\\0&0\end{bmatrix}, ~C=\begin{bmatrix}0&1\\0&0\end{bmatrix}.

    Find all 2x2 matrices A such that AB=BA and AC=CA.

    My analytic solution to this problem was clearly wrong as I got the single zero-matrix as a result. Clearly if A is the identity then those conditions are satisfied, but I don't know the analytic way to solve it correctly.
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  2. #2
    Moo
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    Hello,

    Well, the brute force approach :

    A=\begin{pmatrix}a&b\\c&d\end{pmatrix}

    By performing AB=BA, we get that c,b=0
    By performing AC=CA, we get that c=0,a=d

    So A is in the form \begin{pmatrix}a&0\\0&a\end{pmatrix}=aI_2

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Other thoughts :
    BC=C and CB=0_2

    AB=BA \Rightarrow ABC=BAC \Rightarrow AC=BAC \Rightarrow A=BA (because C is not the zero matrix)


    Maybe it can be done with more abstract algebra (commutative things... ), but I don't know much of it.
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  3. #3
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    Thanks alot, that helps.
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