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Matrix polynomials and inverses- Linear Algebra

  1. Sep 16, 2011 #1
    1. The problem statement, all variables and given/known data
    For p(x)=x4-2x3+3x2-3x+1 and

    A= 1 1 1 -1
    -1 0 -2 1
    0 0 1 0
    1 0 0 0

    you can check that P(A)=0 using this find a polynomial q(x) so that q(A)=A-1. The point is A4-2A3+3A2-3A=A(-A3+2A2-3A+3I)=I

    a) What is q(x)?

    I don't really understand how to approach this problem. My initial though was that I had to solve the right side of the eqn(A4-2A3+3A2-3A) and that would be q(x). Am I on the right track? Also what's the purpose of p(x)?
     
    Last edited: Sep 16, 2011
  2. jcsd
  3. Sep 16, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Well, you say, or were given, that "the point is [itex]A(-A^3+2A^2- 3A+ 3I)= I[/itex]

    Isn't the definition of "inverse" that [itex]AA^{-1}= I[/itex]?
     
  4. Sep 16, 2011 #3

    Mark44

    Staff: Mentor

    Try to be more consistent with your letters. The functions you're working with are named p and q, so you shouldn't be using P in place of p. p(x) = x4-2x3+3x2-3x+1, so p(A) = A4-2A3+3A2-3A+I. Don't worry about the purpose of p - just take it as given for now.

    You have A(-A3+2A2-3A+3I) = I. If A times whatever is the identity, then the whatever is the inverse of A. q(A) will be the inverse of A.
     
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