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N x x determinant problem

  1. Nov 3, 2015 #1
    1. Given [itex]A,B\in Mat _n(\mathbb{R})[/itex]

    2. Show that:
    a) [itex]\det (A^2 + A + E)\geq 0[/itex]
    b) [itex]\det (E+A+B+A^2+B^2)\geq 0[/itex] ,
    where [itex]E[/itex] is the unit matrix.



    3. My attempt at a solution
    [itex]A^2 + A + E[/itex]=[itex](A + E)^2 -2A[/itex]


    https://drive.google.com/file/d/0B8zKPTh1siSsOHNWQnBfaXR3QXM/view?usp=sharing
    Snapshot.jpg
    pleas give me tips to solve
     
    Last edited by a moderator: Nov 3, 2015
  2. jcsd
  3. Nov 3, 2015 #2

    mfb

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    Staff: Mentor

    Something went wrong with the linear term, and I would choose a different term to square.

    You can consider the cases ##det(A)=0## and ##det(A) \neq 0## separately, that gives more freedom to manipulate A in one case.
     
  4. Nov 3, 2015 #3
    Thanks, yes, sorry not - 2A only -A , but than?
     
  5. Nov 3, 2015 #4

    mfb

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    Staff: Mentor

    I would choose a different term to square. A term that doesn't leave an A outside.
     
  6. Nov 3, 2015 #5
    O Yeah!.. I think I found it.. Cayley Hamilton's context A2 - Tr(A)*A+det(A)*E = O
     
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