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Math Help - Matrices

  1. #1
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    Matrices

    Doing some last minute homework, and stuck on this one. Although its due in 4 hours thought its worth a shot.

    Our assignment is to "Use a property of determinants to show that A and A^T have the same characteristic polynomial."

    I know that (A - LI)x = 0, where L = lambda.

    Not sure how to prove that with something like

    det(A-LI) = 0? No idea.

    Also, one other: Show that if A and B are similar, then det A = det B
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Ideasman View Post
    Doing some last minute homework, and stuck on this one. Although its due in 4 hours thought its worth a shot.

    Our assignment is to "Use a property of determinants to show that A and A^T have the same characteristic polynomial."

    I know that (A - LI)x = 0, where L = lambda.

    Not sure how to prove that with something like

    det(A-LI) = 0? No idea.

    Also, one other: Show that if A and B are similar, then det A = det B
    Put B=A-LI, now det(B)=det(B'), so:

    det(B')=det(A'+LI')=det(A'+LI)=det(B)=det(A+LI)

    So the charateristic polynomial of A' (which is det(A'+LI)) is equal to
    det(B'), which is equal to det(B) which is the charateristic polynomial of A
    (that is: det(A-LI)).

    RonL
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