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Question regarding matrix multiplication

  1. May 17, 2005 #1

    When one is trying to multiply two matrices of different sizes e.g. a 2x3 and a 3x3. I know that one has to use the column-row-rule which states:

    [tex]AB_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \cdots + a_{i n} b_{m j} [/tex]

    Looking at the following example:

    [tex] A= \left[ \begin{array}{ccc} 7 & 2 & 0 \\ -4 & 6 & 2 \\ -1 & \frac{2}{5} & \frac{7}{5} \end{array} \right] \ \ \ B = \left[ \begin{array}{ccc} 7 & 2 & 0 \\ -4 & 6 & 2 \\ \end{array} \right][/tex]

    Using the column-row-rule I calculate the matrix-product AB:

    [tex] AB= \left[ \begin{array}{ccc} 7 & 2 & 0 \\ \\ \\ \end{array} \right] \cdot \left[ \begin{array}{cc} -4 &7 \\ 6 &2 \\ 2 & 0 \end{array} \right] = \left[ \begin{array}{cc} -16 & 53 \\ \\ \\ \end{array} \right] [/tex]

    But if I then write the B-matrix upside-down I get:

    [tex] AB= \left[ \begin{array}{ccc} 7 & 2 & 0 \\ \\ \\ \end{array} \right] \cdot \left[ \begin{array}{cc} 0 &2 \\ 2 & 6 \\ 7 & -4 \end{array} \right] = \left[ \begin{array}{cc} 4 & 30 \\ \\ \\ \end{array} \right] [/tex]

    Which of the two results is the correct approach to compute the matrix-product AB ???

    Does there exist a rule in linear algebra which allows me to predetermain if the product of two matrices A and B both not of the same size ( A is n x n and B is m x n ) gives the resulting matrix C which has a different size than A and B ???

    Many thanks in advance :smile:

    Last edited: May 17, 2005
  2. jcsd
  3. May 17, 2005 #2
    you can't do A*B, you can do B*A but A*B is impossible.

    its pretty easy to remember 2x3*3x3 is a 2x3 (just look at the outer numbers, also the inner ones must be the same to be able to multiplicate em)
  4. May 17, 2005 #3
    Okay then the operation I did by tilting the matrix B is illegal. SORRY.

    Then the rows of the matrices A and B has to be equal in-order for the matrix-product AB to be legal??

    In general terms I guess that implies if a matrix A is n x n and a matrix B is m x n then the matrix-product AB is m x n ???

    But there isn't a rule/theorem which details this??



  5. May 17, 2005 #4
    If a matrix A is n x n and a matrix B is m x n then the matrix-product AB does't exist at all. You can compute only BA.
    In general. If A is m x k, and B is k x n, than AB is m x n. But BA isn't defined. Number of columns of first matrix must be equal to number of rows of second
  6. May 17, 2005 #5

    I think you're still confused.

    For A*B to be defined, the # of columns in A has to equal the # of rows in B. So A(j,k)*B(k,m) = C(j,m). But A(j,k)*B(m,n) is not defined unless k=m.

    No theory is really required for this; it's just the definition of matrix multiplication.
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