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    The characteristic equation has at most $n$ roots in $F$

    E-HK-54-10

    Area
    Algebra
    Topics
    Characteristic Values, Determinants, Linear Algebra
    Status
    draft

    Let \(A\) be an \(n \times n\) matrix over the field \(F\). Prove that there are at most \(n\) distinct scalars \(c\) in \(F\) such that \(\det(cI - A) = 0\).

    Solution

    The function \[ p(x)=\det(xI-A) \] is the characteristic polynomial of \(A\). It is a monic polynomial of degree \(n\), hence is nonzero. Over a field, a nonzero polynomial of degree \(n\) has at most \(n\) distinct roots. Therefore there are at most \(n\) scalars \(c\in F\) satisfying \[ \det(cI-A)=p(c)=0. \]

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