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Math Help - Matrices & Systems of Equations:Matrix Algebra

  1. #1
    Junior Member
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    May 2007
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    Matrices & Systems of Equations:Matrix Algebra

    Hello could someone please help me with the following problems, I just couldn't find my way around it. Thanks!

    1) Given the following:
    A equals
    1 2
    1 -2

    b equals
    4
    0

    c equals
    -3
    -2


    write c as a linear combination of column vectors a_1 & a_2

    2) Let A equals

    a_11 a_12
    a_21 a_22

    Show that if d = a_11 a_22 - a_21 a_12 is not equal 0, then
    A inverse equals

    1/d * ( {a_22 -a_12} {-a_21 a_11} ) ====> {row 1} {row 2}
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by googoogaga View Post
    Hello could someone please help me with the following problems, I just couldn't find my way around it. Thanks!

    1) Given the following:
    A equals
    1 2
    1 -2

    b equals
    4
    0

    c equals
    -3
    -2


    write c as a linear combination of column vectors a_1 & a_2
    Suppose x~\bold{a_1}+y ~\bold{a_2}=\bold{c}, then we may write this as a system of linear equations:

    <br />
x+2y=-3<br />
    <br />
x-2y=-2<br />

    which you can now solve for x and y.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by googoogaga View Post
    2) Let A equals

    a_11 a_12
    a_21 a_22

    Show that if d = a_11 a_22 - a_21 a_12 is not equal 0, then
    A inverse equals

    1/d * ( {a_22 -a_12} {-a_21 a_11} ) ====> {row 1} {row 2}
    This requires that you multiply the matrices:

    <br />
\left( \begin{array}{cc} a_{11}&a_{12}\\a_{21}&a_{22} \end{array} \right)<br />

    and:

    <br />
\frac{1}{d}\left( \begin{array}{cc} a_{22}&-a_{12}\\-a_{21}&a_{11} \end{array} \right)<br />

    to show that this is the 2x2 identity matrix

    RonL
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  4. #4
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    Matrices and Systems Equations

    I still don't get number two. Once multiplying the matrices, then where do I go from there? And what is the 2*2 identity matrice?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by googoogaga View Post
    I still don't get number two. Once multiplying the matrices, then where do I go from there? And what is the 2*2 identity matrice?
    You need to show that
    \frac{1}{a_{11}a_{22} - a_{12}a_{21}}\left ( \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{matrix} \right ) \cdot \left ( \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} \right ) = \left ( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right ) = I_2

    -Dan
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