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    Polynomial and matrix

    Suppose there is a matrix A\in\mathbb{Q} that satisfies the polynomial x^{3}+x^{2}-x-1. Prove that 3 divides n.
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    Re: Polynomial and matrix

    Quote Originally Posted by kierkegaard View Post
    Suppose there is a matrix A\in\mathbb{Q} that satisfies the polynomial x^{3}+x^{2}-x-1. Prove that 3 divides n.
    What is n?
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    Re: Polynomial and matrix

    Quote Originally Posted by kierkegaard View Post
    Suppose there is a matrix A\in\mathbb{Q} that satisfies the polynomial x^{3}+x^{2}-x-1. Prove that 3 divides n.
    1) There is no "n" in the given information.
    2) It makes no sense to say that something "satisfies a polynomial". A number or matrix may satisfy a polynomial equation but you have no equation here. Do you mean "satisfies the polynomial equation x^2+ x^2- x- 1= 0"?
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    Re: Polynomial and matrix

    This is a corrected answer. Thanks.

    Suppose there is a matrix A\inM_{n}(\mathbb{Q}) that satisfies the polynomial x^{3}+x^{2}-x+1=0. Prove that 3 divides n.
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    Re: Polynomial and matrix

    Quote Originally Posted by kierkegaard View Post
    Suppose there is a matrix A\in M_{n}(\mathbb{Q}) that satisfies the polynomial x^{3}+x^{2}-x+1=0. Prove that 3 divides n.
    If A^3+A^2-A+I = 0 then the minimum polynomial of A must be a factor of p(x) = x^{3}+x^{2}-x+1. But that polynomial is irreducible over the rationals. Therefore p(x) must be the minimum polynomial of A. The factors of the characteristic polynomial of A must be the same as those of the minimum polynomial. But again, since p(x) is irreducible over \mathbb{Q}, it only has one factor, namely itself. Therefore the characteristic polynomial must be of the form p(x)^k for some integer k. This has degree 3k, and so A\in M_{3k}(\mathbb{Q}), as required.

    That naturally raises the question of whether there are any such matrices. In fact there are, and you can check that B = \begin{bmatrix}1&1&0 \\ -1&-1&1 \\ 0&1&-1 \end{bmatrix} satisfies p(B)=0. The most general 3x3 matrix satisfying the equation would then be similar to B. In other words, it would be of the form P^{-1}BP, where P is any invertible matrix in M_{3}(\mathbb{Q}). In general, any matrix satisfying the equation would be similar to a direct sum of copies of B.
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