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Multiplication of matrices properties

  1. Nov 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let ej denote the jth unit column that contains a 1 in the jth
    position and zeros everywhere else. For a general matrix An×n, describe the following products. (a) Aej (c) eTiAej?

    2. Relevant equations
    Rows and Columns of a Product
    Suppose that A = [aij] is m × p and B = [bij] is p × n.
    • [AB]i∗ = Ai∗B [( ith row of AB)=( ith row of A) ×B]. (3.5.4)

    • [AB]∗j = AB∗j [ (jth col of AB)=A× ( jth col of B)]. (3.5.5)

    • [AB]i∗ = ai1B1∗ + ai2B2∗ + · · · +aipBp∗aikBk∗. (3.5.6)

    • [AB]∗j = A∗1b1j + A∗2b2j + · · · + A∗pbpjA∗kbkj (3.5.7)

    These last two equations show that rows of AB are combinations of
    rows of B, while columns of AB are combinations of columns of A.


    3. The attempt at a solution
    For parts a and c im not even sure what they are even asking for. When its saying ej is a unit column does that mean like this (1 0 0...0) as an example? For part A wouldn't the solution of Aej just just be a linear combination a column of A and the entries of ej as scalars?
     
    Last edited: Nov 18, 2012
  2. jcsd
  3. Nov 18, 2012 #2

    haruspex

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    Yes, it means that ej = (δij)T, i.e. 1 where i=j and zero elsewhere.
    From your matrix multiplication formulae, can you deduce a formula for B being a column vector instead of a matrix?
     
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