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Ring of matrix over a field

  1. Jan 30, 2010 #1
    Let R = Mn(F) the ring consists of all n*n matrices over a finite field F and

    E= E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix(Eij is matrix whose ij th element is 1 and the others are 0). Then the following hold:

    1. If A is a rank n-1 matrix in RE then A is similar to E.

    what is the proof of the above statement?
    thank you
     
  2. jcsd
  3. Feb 4, 2010 #2

    radou

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    You need to prove that any matrix of rank n-1 can be transformed to a similar elementary matrix of rank n-1 by applying a finite number of elementary row/column transformations.
     
  4. Feb 4, 2010 #3
    for similarity between A and E there must be an invertible matrix P such that [tex]\textit{P}[/tex][tex]\textit{A}[/tex][tex]\textit{\textit{P}}^{-1}[/tex][tex]\textit{=E}[/tex]
    how can I say that?
     
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