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

5p a) Arătați că det [tex]$A=-5$[/tex][

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

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

[tex]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)[/tex]

[tex]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)[/tex]

Se observa ca sunt egale

c)

A·A-3(A+M(x))=I₂

[tex]\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)[/tex]

[tex]\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)[/tex]

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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