Se consideră matricele [tex]$A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)$[/tex] şi [tex]$I_{3}=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$[/tex].

5p a) Arătați că [tex]$\operatorname{det}\left(A+I_{3}\right)=4$[/tex].

[tex]$5 p$[/tex] b) Demonstrați că [tex]$A \cdot A \cdot A+A=2 A \cdot A$[/tex].

[tex]$5 p$[/tex] c) Determinați mulțimea valorilor reale ale lui [tex]$x$[/tex] pentru care matricea [tex]$B(x)=A+x I_{3}$[/tex] este inversabilă.

[tex]A\times A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)\times \left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 3& 1\end{array}\right)[/tex]

[tex]A\times A\times A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 3& 1\end{array}\right) \times \left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1& 1\end{array}\right) =\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 5& 1\end{array}\right)[/tex]

[tex]A\times A\times A+A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 5& 1\end{array}\right) +\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1& 1\end{array}\right) =\left(\begin{array}{lll}0 & 2 & 0 \\ 0 & 2 & 0 \\ 2 & 6& 2\end{array}\right) =2\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 3& 1\end{array}\right) =2A\times A[/tex]

c)

Daca B(x) este inversabila, atunci det(B(x))≠0

[tex]det(B(x))=\left|\begin{array}{ccc}x&1&0\\0&x+1&0\\1&1&x+1\end{array}\right|[/tex]

x 1 0

0 x+1 0

det(B(x))=[x(x+1)²+0+0]-0=x(x+1)²

x(x+1)²≠0

x≠0 si x≠-1

x∈R\{-1,0}

Un alt exercitiu similar de bac il gasesti aici: https://brainly.ro/tema/3845583

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