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Linear Algebra- Matrices

  1. May 21, 2008 #1
    1. The problem statement, all variables and given/known data

    You are given matrices A:

    A=
    [-4 5 9 0 3
    -5 1 3 8 -5
    -6 0 4 0 -9
    -1 -1 -4 3 -5]
    and

    B=
    [-4 5 9 0 3
    -6 0 4 0 -9
    -40 8 24 64 -40
    -1 -1 -4 3 -5 ]

    Find two elementary matrices E and F that transform matrix A into matrix B.



    2. Relevant equations

    The solution might not be unique, however, after multiplying A on the left with elementary matrices E and F your result should be matrix B. That is, F E A = B.


    3. The attempt at a solution

    I know one Elementary matrix already which is R2*8
    and the second elementary matrix swaps row 2 with row 3


    do I write it like this:
    F=
    [1 0 0 0 0
    0 8 0 0 0
    0 0 1 0 0
    0 0 0 1 0] --is it okay to have the last column full of zeros??

    and the next elementary matrix, do i write it like this:
    [1 0 0 0 0
    0 0 1 0 0
    0 1 0 0 0
    0 0 0 1 0]
     
    Last edited: May 21, 2008
  2. jcsd
  3. May 21, 2008 #2
    I think elementary matrices need to be square matrices ...

    think about 4x4 matrices

    edit 2: (undoing edit 1 lol): can you multiply those matrices?!
    5x4 * 5x4..
     
    Last edited: May 21, 2008
  4. May 22, 2008 #3

    Defennder

    User Avatar
    Homework Helper

    Note that elementary matrices are invertible, so they have to be square matrices. Note that a 4x4 matrix multiplied to a 4x5 matrix will give you a 4x5 matrix, which is what you want.

    As you pointed out, you need 2 matrices, one which reflects 8*R2 and one which swaps R2, R3. Note that the elementary matrix which performs a desired row operation when multiplied to another matrix is obtained by executing the same row operation on the identity matrix.
     
  5. May 22, 2008 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    An elementary matrix is a matrix created from the identity matrix by a single "row operation". I see that A and B differ in only two ways: The second and third rows have been swapped and the (new) third row is multiplied by 8. Apply those two operations to the identity matrix to get the two elementary matrices required.
     
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