Question

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

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

$5 p$ b) Demonstrați că $A \cdot A \cdot A+A=2 A \cdot A$.

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

Answer 1

$A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)$

a)

det(A+I₃)=4

Calculam det(A+I₃)

$det(A+I_3)=A=\left|\begin{array}{lll}1 & 1 & 0 \\ 0 &2 & 0 \\ 1 & 1 & 2\end{array}\right|$

                                1    1    0

                                0   2   0

det(A+I₃)=(4+0+0)-(0+0+0)=4

b)

$A\times A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)\times \left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 1\end{array}\right)=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 3& 1\end{array}\right)$

$A\times A\times A=\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 3& 1\end{array}\right) \times \left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1& 1\end{array}\right) =\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 5& 1\end{array}\right)$

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

c)

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

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

                      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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