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Math Help - finding characteristic polynomial, problem with the question?

  1. #1
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    finding characteristic polynomial, problem with the question?

    Hi.
    I was asked to find the characteristic polynomial of A.
    A has the following attributes:

    A\in M_n(F)

    rankA=1

    n\geq 2

    traceA=0

    I argue that there is a contradiction here!

    if rankA=1, it means that: dimV_0=dimKerA=n-1.
    if dimKerA=n-1 then it means that the diagonal matrix D (who has the eignvalues on it's main diagonal), has some \lambda in its d_{1,1} entry, and the rest are zeroes. meaning, D looks like this:
    \begin{bmatrix}\lambda & 0 & . & . & . & 0\\ 0 & 0 &  &  &  & .\\ . &  & 0 &  &  & .\\ . &  &  & 0 &  & .\\ . &  &  &  & 0 & .\\ 0 & 0 & . & . & . & 0\end{bmatrix}
    now, the problem is, that traceA=0, and since the trace is the sum of the eignvalues, we'll get that \lambda=0, and that means that rankA=0! (and not 1 as stated in the question)
    Last edited by Stormey; April 26th 2013 at 02:11 AM.
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  2. #2
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    Re: finding characteristic polynomial, problem with the question?

    Hi,
    I think you'd best rethink your analysis. Does the following matrix satisfy the conditions?

    A=\begin{bmatrix}1&1\\-1&-1\end{bmatrix}

    What is the characteristic polynomial of A?
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  3. #3
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    Re: finding characteristic polynomial, problem with the question?

    OK, now I see what was wrong with my analysis.
    I thought that elementary row operations doesn't change the transformation.
    dunno why...
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