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Matrices proof.

  1. Sep 20, 2011 #1
    1. The problem statement, all variables and given/known data
    28. Let A be an m x n matrix with a row consisting entirely of zeros. Show that if B is an n x p matrix, then BA has a row of zeros.

    2. Relevant equations
    N/A


    3. The attempt at a solution
    [itex]A = (aij)_{mxn}[/itex] and [itex]B = (bij)_{nxp}[/itex]. Assuming that the entries for jth column of A are all zeros, I infer that the jth column of BA is also a row of zeros. Let [itex]cij[/itex] be an entry in jth column. By the definition of multiplication:
    [tex] cij = b_{i1}a_{j1} + b_{i2}a_{j2} + ...+ b_{in}a_{nj} = \sum_{k=1}^n b_{ik}a_{kj}[/tex]

    Since the jth column of A is zero, then there is:
    [itex]a_{1j} = a_{2j} = ... = a_{nj} = 0[/itex]. Hence, [itex]cij = 0[/itex] and, therefore, the jth column of BA is a column of zeros.

    Did I do this proof correctly? Thanks in advance.
     
    Last edited: Sep 20, 2011
  2. jcsd
  3. Sep 21, 2011 #2

    Mark44

    Staff: Mentor

    You're given that one row of A consists of zeros. You can't also assume that one column (column j) is all zeros.
     
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