Ivan Shishkin, Rye (1878)

Problems/OtherExerciseUnreviewed

Polynome caractéristique d’une simple matrice

by SalixBabylonica·
20
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  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
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On considère la matrice A=(1234)A =\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.

  1. Calculer son polynôme caractéristique.
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Solution by SalixBabylonica

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Par définition, le polynôme caractéristique est donné par

χA(X)=det⁡(XI2−A).\chi_A(X) = \det(X I_2 - A).

Commençons par former la matrice XI2−AX I_2 - A :

XI2−A=(X−1−2−3X−4).X I_2 - A = \begin{pmatrix} X - 1 & -2 \\ -3 & X - 4 \end{pmatrix}.

Le déterminant de cette matrice s’écrit alors

χA(X)=(X−1)(X−4)−(−2)(−3)=X2−5X+4−6=X2−5X−2.\chi_A(X) = (X - 1)(X - 4) - (-2)(-3) = X^2 - 5X + 4 - 6 = X^2 - 5X - 2.

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