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Math Help - Proving properties of matrix

  1. #1
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    Proving properties of matrix

    Let A be the matrix \left(\begin{array}{cc}a&b\\c&d\end{array}\right), where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
    (a) a+d=0, in which case the general solution for X depends on two parameters, or
    (b) ad-bc=0, in which case the general solution for X depends on one parameter.

    I don't know where to begin other than naming X= \left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{  22}\end{array}\right) and then find the matrices AX and XA.
    Thanks
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  2. #2
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    Quote Originally Posted by arze View Post
    Let A be the matrix \left(\begin{array}{cc}a&b\\c&d\end{array}\right), where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
    (a) a+d=0, in which case the general solution for X depends on two parameters, or
    (b) ad-bc=0, in which case the general solution for X depends on one parameter.

    I don't know where to begin other than naming X= \left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{  22}\end{array}\right) and then find the matrices AX and XA.
    Thanks

    Exactly, that's what you have to do...and then solve AX+XA=0 , where the zero in the right is, of course, the zero 2x2 matrix.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Exactly, that's what you have to do...and then solve AX+XA=0 , where the zero in the right is, of course, the zero 2x2 matrix.

    Tonio
    Ok, i did that and this is what I got

    AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1  2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e  nd{array}\right)
    XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1  1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e  nd{array}\right)
    equating it with the zero matrix I get four equations.
    2ax_{11}+x_{12}(b+c)=0 __1
    b(x_{22}+x_{11})+x_{12}(a+d)=0 __2
    x_{21}(a+d)+c(x_{11}+x_{22})=0 __3
    2dx_{22}+cx_{12}+bx_{21}=0 __4

    Now what do I do next?
    Thanks
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  4. #4
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    [quote=arze;484206]Ok, i did that and this is what I got

    AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1  2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e  nd{array}\right)
    XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1  1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e  nd{array}\right)
    equating it with the zero matrix I get four equations.
    2ax_{11}+x_{12}(b+c)=0 __1
    b(x_{22}+x_{11})+x_{12}(a+d)=0 __2
    x_{21}(a+d)+c(x_{11}+x_{22})=0 __3
    2dx_{22}+cx_{12}+bx_{21}=0 __4

    Now what do I do next?


    Solve the system of equations, what else?! But first fix the first one: it must be

    2ax_{11}+bx_{21}+cx_{12}=0

    Tonio
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  5. #5
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    Oops! my bad
    I have tried but can't seem to get them.
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  6. #6
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    Quote Originally Posted by arze View Post
    Oops! my bad
    I have tried but can't seem to get them.

    You have to work harder and make a bigger effort. The matrix of coefficients of the system ( in the unknonws x_{11},x_{12},x_{21},x_{22} ) is:

    \begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c  &b&2d\end{pmatrix} . Divide now the first row by 2a (why can you?) and substract from each of the 2nd, 3rd rows a corresponding multiple of the first one to obtain:

    \begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&a+d-bc\slash 2a&-b^2\slash 2a&b\\0&-x^2\slash 2a&a+d-bc\slash 2a&c\\0&c&b&2d\end{pmatrix} . Interchange now the 2nd and 4th rows, divide the new 2nd row by c (again, why can you) and ...etc. This is

    just the Gauss-Jordan reduction method of a matrix to echelon form!:

    \begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&1&b\slash c&2d\slash c\\0&0&-\frac{b}{c}(a+d)&\left(\frac{b}{a}-\frac{2d}{c}\right)(a+d)\\0&0&a+d&\frac{c}{a}(a+d)  \end{pmatrix} ...or something like this (check this carefully: I didn't!) ...and etc.

    Tonio
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  7. #7
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    Quote Originally Posted by tonio View Post

    \begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c  &b&2d\end{pmatrix} .
    I don't understand this part. How did you get that?
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