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Homework Help: Given a Matrix A, find a Product of Elementary Matrices that equals A

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Given

    [tex]A[/tex] = [tex]\left( \begin{array}{cc}
    2 & 1 \\
    6 & 4 \end{array} \right)[/tex]

    a) Express A as a product of elementary matrices.
    b) Express the inverse of A as a product of elementary matrices.

    2. Relevant equations



    3. The attempt at a solution

    Using the following EROs

    Row2 --> Row2 - 3 * Row1
    [tex]E_{1}[/tex] = [tex]\left( \begin{array}{cc}
    1 & 0 \\
    -3 & 1 \end{array} \right)[/tex]

    Row1 --> 1/2 * Row1
    [tex]E_{2}[/tex] = [tex]\left( \begin{array}{cc}
    1/2 & 0 \\
    0 & 1 \end{array} \right)[/tex]

    Row1 --> Row1 - 1/2 * Row2
    [tex]E_{3}[/tex] = [tex]\left( \begin{array}{cc}
    1 & 0 \\
    -2 & 1 \end{array} \right)[/tex]

    Multiplying all the Elementary matrices together, I got the Product

    [tex]P[/tex] = [tex]\left( \begin{array}{cc}
    2 & -1/2 \\
    -3 & 1 \end{array} \right)[/tex]

    Which is A-1.
     
    Last edited: Feb 4, 2010
  2. jcsd
  3. Feb 4, 2010 #2
    Note that the inverse of A-1 is A and also that given invertible A and B, (AB)-1=B-1A-1

    You have E1E2....En=A-1 where Ei is an elementary matrix. So take the inverse of the whole thing
     
  4. Feb 4, 2010 #3
    If I'm understanding you correctly, I should take the inverses of all the elementary matrices and multiply those, and it should give me A?

    Essentially, (E1E2....En)-1 = A
     
    Last edited: Feb 4, 2010
  5. Feb 4, 2010 #4
    Hey Sandwich,

    Think of the matrix A as being equivalent to an identity matrix of the same size, but just manipulated by elementary row operations.

    Vee is right, because if you multiply the inverse of A by A's corresponding elementary matrices, the product is the identity matrix. Try it out.

    A=I(E1E2....En)^(-1)

    Hope that helps!
     
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