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Math Help - Matrices- Trace

  1. #1
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    Matrices- Trace

    show that for all A,B nxn matrices over the field F:
    a)trace(X+Y)=trace(X) + trace(Y)
    b)trace(XY)=trace(YX)
    c)show that there are no nxn matrices X,Y such that XY-YX= indentity martrix
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  2. #2
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    Hi amm345,
    Quote Originally Posted by amm345 View Post
    show that for all A,B nxn matrices over the field F:
    a)trace(X+Y)=trace(X) + trace(Y)
    b)trace(XY)=trace(YX)
    c)show that there are no nxn matrices X,Y such that XY-YX= indentity martrix
    a) follows from the definition of matrix addition, tr(A+B) = \sum_{i=1}^n a_{ii}+b_{ii} = \sum_{i=1}^n a_{ii}+\sum_{i=1}^n b_{ii} = tr(A)+tr(B)

    b) uses commutativity of the field, tr(AB)= \sum_{i=1}^n \sum_{j=1}^n a_{ij} b_{ji} =\sum_{j=1}^n \sum_{i=1}^n b_{ji}a_{ij}= tr(BA)

    c) follows from b)
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