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Math Help - show for a if a square matrix A...it will hold for A^T

  1. #1
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    show for a if a square matrix A...it will hold for A^T

    Show that if for a square matrix $A$ satisfies $A^3+ 4A^2 - 2A + 7I = 0 $ then so does $A^T$

    I suspect that this is not always true so I need to do an example not the general case ( I partially suspect this cause the next question asks me if this is always true or sometimes false so I need to find a counter-example as well)

    not really sure where to start this problem.
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    Re: show for a if a square matrix A...it will hold for A^T

    Quote Originally Posted by Jonroberts74 View Post
    Show that if for a square matrix $A$ satisfies $A^3+ 4A^2 - 2A + 7I = 0 $ then so does $A^T$

    I suspect that this is not always true so I need to do an example not the general case ( I partially suspect this cause the next question asks me if this is always true or sometimes false so I need to find a counter-example as well)

    not really sure where to start this problem.
    $\left(A^T\right)^2=\left(A^2\right)^T$

    $\left(A^T\right)^3=\left(A^3\right)^T$

    $(A^3+4A^2-2A+7I)^T=0^T=0$

    $(A^3)^T+(4A^2)^T-(2A)^T+(7I)^T=0$

    $(A^T)^3+4(A^T)^2-2(A^T)+7I=0$

    so $A^T$ satisfies the equation as well.
    Thanks from Jonroberts74
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    Re: show for a if a square matrix A...it will hold for A^T

    ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

    Does A have to be symmetric for this to work?
    Last edited by Jonroberts74; September 2nd 2014 at 12:26 AM.
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    Re: show for a if a square matrix A...it will hold for A^T

    Quote Originally Posted by Jonroberts74 View Post
    ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

    Does A have to be symmetric for this to work?
    You know that \left(AB\right)^T = B^TA^T. Just set A = B.
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    Re: show for a if a square matrix A...it will hold for A^T

    Quote Originally Posted by Jonroberts74 View Post
    ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?

    Does A have to be symmetric for this to work?
    no

    $(A\cdot A \cdot A \dots)^T=A^T\cdot A^T \cdot A^T \dots$

    so

    $(A^n)^T=(A^T)^n$
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