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Modifications apportées au TD2 de mathématiques générales

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Feror 2026-04-02 11:06:07 +02:00
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MathématiquesGénérales/TD2.md
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![[CI-SST81E6_TD2.pdf]]
# Exercice 1: # Exercice 1:
$A=\pmatrix{-1&0&1\\1&0&-1\\0&0&0}$ $A=\pmatrix{-1&0&1\\1&0&-1\\0&0&0}$
## 1) ## 1)
$X_A(x)=\det{(x\times I - A)}=\matrix{+\\-\\+}\begin{vmatrix}x+1&0&-1\\-1&x&1\\0&0&x\end{vmatrix}=x^3+x^x=x^2(x+1)$ $X_A(x)=\det{(x\times I - A)}=\matrix{+\\-\\+}\begin{vmatrix}x+1&0&-1\\-1&x&1\\0&0&x\end{vmatrix}=x^3+x^x=x^2(x+1)$
## 2) ## 2)
$A\times (A+I)$ On veut calculer $A\times (A+I)$
$A+I=\pmatrix{0&0&1\\1&1&1\\0&0&1}\pmatrix{-1&0&1\\1&0&-1\\0&0&0}=\pmatrix{0&0&0\\0&0&0\\0&0&0}$ $A+I=\pmatrix{0&0&1\\1&1&-1\\0&0&1}$
$A\times (A+I)=\pmatrix{0&0&1\\1&1&-1\\0&0&1}\pmatrix{-1&0&1\\1&0&-1\\0&0&0}=\pmatrix{0&0&0\\0&0&0\\0&0&0}$
$p:x\rightarrow x(x+1)$. $p:x\rightarrow x(x+1)$.
$p$ est le polynome minimal de $A$. $p$ est le polynome minimal de $A$.