Question

Se consideră matricele $A=\left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right), B=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), I_{2}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ și $M(x)=B+x I_{2}$, unde [$x$ este număr real.

5p a) Arătați că det $A=-5$[

$5 p$ b) Arătați că $A \cdot M(x)=M(x) \cdot A$, pentru orice număr real $x$.

$5 p$ c) Determinaţi numărul real $x$ pentru care $A \cdot A-3(A+M(x))=I_{2}$.

Answer 1

$A=\left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right)$

$B=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$

a)

Calculam detA, facem diferenta dintre produsul diagonalelor

detA=4-9=-5

b)

A·M(x)=M(x)·A

M(x)=B+xI₂

$A\cdot M(x)=\left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right)\cdot \left(\begin{array}{ll}x & 1\\ 1& x\end{array}\right)=\left(\begin{array}{ll}2x+3 &2 +3x\\ 3x+2& 3+2x\end{array}\right)$

$M(x)\cdot A= \left(\begin{array}{ll}x & 1\\ 1& x\end{array}\right)\cdot \left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right)=\left(\begin{array}{ll}2x+3 &2 +3x\\ 3x+2& 3+2x\end{array}\right)$

Se observa ca sunt egale

c)

A·A-3(A+M(x))=I₂

$\left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right)\cdot \left(\begin{array}{ll}2 & 3 \\ 3 & 2\end{array}\right)-3\left(\begin{array}{ll}2+x & 4 \\ 4 & 2+x\end{array}\right)=\left(\begin{array}{ll}1& 0 \\ 0 & 1\end{array}\right)$

$\left(\begin{array}{ll}13 & 12\\ 12 & 13\end{array}\right)-\left(\begin{array}{ll}6+3x & 12 \\ 12 & 6+3x\end{array}\right)=\left(\begin{array}{ll}1& 0 \\ 0 & 1\end{array}\right)$

13-6-3x=1

7-3x=1

6=3x

x=2

Un alt exercitiu cu matrice gasesti aici: https://brainly.ro/tema/9835805

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