10 lines
357 B
Markdown
10 lines
357 B
Markdown
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# Exercice 1:
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$A=\pmatrix{-1&0&1\\1&0&-1\\0&0&0}$
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## 1)
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$X_A(x)=\det{(x\times I - A)}=\matrix{+\\-\\+}\begin{vmatrix}x+1&0&-1\\-1&x&1\\0&0&x\end{vmatrix}=x^3+x^x=x^2(x+1)$
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## 2)
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$A\times (A+I)$
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$A+I=\pmatrix{0&0&1\\1&1&1\\0&0&1}\pmatrix{-1&0&1\\1&0&-1\\0&0&0}=\pmatrix{0&0&0\\0&0&0\\0&0&0}$
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$p:x\rightarrow x(x+1)$.
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$p$ est le polynome minimal de $A$.
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