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Matrix multiplication problem.

  1. Apr 6, 2005 #1
    I know how to solve basic problems like this, but I have no clue where to start with one of the first parts in this example. I am given the following information about C, which is a 2 X 2 matrix.
    [itex]C \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right] = \left[
    \begin{array}{cc}
    2\\
    1
    \end{array}
    \right] [/itex] and [itex]C^2 \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right] = \left[
    \begin{array}{cc}
    -1\\
    1
    \end{array}
    \right] [/itex]

    The question asks for 2 X 2 matrices A and B so that CA = B, then solve for C.

    My problem is finding what the B matrix is. A is [itex]\left[
    \begin{array}{cc}
    1 & 2\\
    2 & 1
    \end{array}
    \right][/itex], so how can I find B if C is squared?
     
  2. jcsd
  3. Apr 7, 2005 #2

    xanthym

    User Avatar
    Science Advisor

    SOLUTION HINTS:
    From the problem statement, we have:

    [tex] 1: \ \ \ \ C \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right] \ = \ \left[
    \begin{array}{cc}
    2\\
    1
    \end{array}
    \right] [/tex]

    [tex] 2: \ \ \ \ C^2 \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right] \ = \ C \cdot C \cdot \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right]
    \ = \ \left[
    \begin{array}{cc}
    -1\\
    1
    \end{array}
    \right] [/tex]

    Thus, the following is also known from Eq #2 (together with Eq #1):

    [tex] 3: \ \ \ \ \color{red} C \cdot \color{blue} \left ( C \cdot \left[
    \begin{array}{cc}
    1\\
    2
    \end{array}
    \right] \right ) \ = \ \color{red} C \cdot \color{blue} \left[
    \begin{array}{cc}
    2\\
    1
    \end{array}
    \right] \ = \ \color{red} \left[
    \begin{array}{cc}
    -1\\
    1
    \end{array}
    \right] [/tex]

    Hence, from Eq #1 & #3, we can now write:

    [tex] 4: \ \ \ \ C \cdot \left[
    \begin{array}{cc}
    1 & 2\\
    2 & 1
    \end{array}
    \right] \ = \ \left[
    \begin{array}{cc}
    2 & -1\\
    1 & 1
    \end{array} \right ] [/tex]

    Solve for "C" by finding the INVERSE of the matrix shown below and multiplying both sides of the last equation (Eq #4) from the RIGHT:

    [tex] 5: \ \ \ \ \left[
    \begin{array}{cc}
    1 & 2\\
    2 & 1
    \end{array}
    \right] [/tex]


    ~~
     
    Last edited: Apr 7, 2005
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