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Homework Help: Find matrices X given an equation.

  1. Jul 22, 2010 #1
    1. The problem statement, all variables and given/known data

    find all matrices x that satisfy the given matrix equation
    [ 1 2 3
    4 5 6] * X = I_2

    I_2 is the identity matrix 2x2
    2. Relevant equations



    3. The attempt at a solution

    I just inverted the square matrix
    [ 1 2
    4 5]
    so it becomes
    [5 -2
    -4 1 ]
    so X should be
    [ 5 -2
    -4 1
    0 0]

    but my book solution introduces 2 variables S and T to capture all the solutions, how do i do this?



    *ps, how do i make my matrices neater?
     
  2. jcsd
  3. Jul 22, 2010 #2

    hunt_mat

    User Avatar
    Homework Helper

    The way you went about it is a little odd. I personally would have written:
    [tex]
    \left(
    \begin{array}{ccc}
    1 & 2 & 3 \\
    4 & 5 & 6
    \end{array}
    \right) \left(
    \begin{array}{cc}
    a_{1} & a_{2} \\
    a_{3} & a_{4} \\
    a_{5} & a_{6}
    \end{array}
    \right) =\left(
    \begin{array}{cc}
    1 & 0 \\
    0 & 1
    \end{array}
    \right)
    [/tex]
    Multiplied them out and solved the linear equations. You will get four equations to solve for 6 unknowns, this is where the variables t & s come into it.
     
  4. Jul 22, 2010 #3
    I would just treat it as two simple linear algebra problems of finding the solution space for Ax=b:


    [tex]

    \left(
    \begin{array}{ccc}
    1 & 2 & 3 \\
    4 & 5 & 6
    \end{array}
    \right) [/tex]

    simplifies to the reduced row-echelon form matrix

    [tex]
    \left(
    \begin{array}{ccc}
    1 & 0 & -1 \\
    0 & 1 & 2
    \end{array}
    \right) [/tex]

    So we have
    [tex]

    \left(
    \begin{array}{ccc}
    1 & 0 & -1 \\
    0 & 1 & 2
    \end{array}
    \right) \left(
    \begin{array}{c}
    a_{1} \\
    a_{3} \\
    a_{5}
    \end{array}
    \right) =\left(
    \begin{array}{c}
    1 \\
    0
    \end{array}
    \right)

    [/tex]

    and
    [tex]


    \left(
    \begin{array}{ccc}
    1 & 0 & -1 \\
    0 & 1 & 2
    \end{array}
    \right) \left(
    \begin{array}{c}
    a_{2} \\
    a_{4} \\
    a_{6}
    \end{array}
    \right) =\left(
    \begin{array}{c}
    0 \\
    1
    \end{array}
    \right)

    [/tex]
     
  5. Jul 22, 2010 #4
    thanks for the replies.
    I see how it's more intuitive to do it the algebraic way, so I'll just discard the inverse trick.
    Using the rref makes it pretty simple, but when i reduce the one on the left side, do I reduce it on the right as well, if say we're given a matrix other than an I_n ?
     
  6. Jul 22, 2010 #5
    Yea, sorry I forgot to change the right side. But yep, just apply Gauss-Jordan elimination to the augmented matrix (which includes the right-hand side).
     
  7. Jul 22, 2010 #6
    can anyone verify this?

    my 1st column came out to be
    a = t
    b = -2 -2t
    c = 5/3 +t

    and 2nd column
    d= s
    e= 1-2s
    f= -2/3+s
     
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