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Proving the scalar matrices are the center of the matrix ring

  1. Sep 12, 2013 #1
    I read that scalar matrices are the center of the ring of matrices. How would I prove this?
    Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky.

    BiP
     
  2. jcsd
  3. Sep 12, 2013 #2
    Supposing R is commutative with identity:

    Let C be in the center of M(R). If C is not diagonal then cij≠0 for some i≠j. Letting A be the matrix with 1 in the ji position and 0 elsewhere, we see that CA≠AC, contradicting the fact that C is in the center of M(R). So C is diagonal. If C is diagonal but cii≠cjj, then we can again find a matrix in M(R) which does not commute with C.
     
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