Question

Se consideră matricele $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right), B=\left(\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right)$ şi $C=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$.

5p a) Arătați că det $A=-2$.

$5 p$ b) Determinaţi numărul real $x$ pentru care $x(A+B)=C$.

$5 p$ c) Determinați matricea $X \in \mathcal{M}_{2}(\mathbb{R})$ pentru care $A \cdot B-B \cdot A=2 X+C$.

Answer 1

$A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$

$B=\left(\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right)$

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

a)

Facem diferenta dintre produsul diagonalelor

detA=4-6=-2

b)

$A+B=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)+\left(\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right)=\left(\begin{array}{ll}5 & 5 \\ 5 & 5\end{array}\right)=5C$

$5xC=C\\\\5x=1\\\\x=\frac{1}{5}$

c)

$A\cdot B=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)\cdot \left(\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right)=\left(\begin{array}{ll}8 & 5 \\ 20 & 13\end{array}\right)$

$B\cdot A= \left(\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right)\cdot \left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)=\left(\begin{array}{ll}13 & 20 \\ 5 & 8\end{array}\right)$

$A\cdot B-B\cdot A=\left(\begin{array}{ll}-5& -15 \\ 15 & 5\end{array}\right)\\\\\left(\begin{array}{ll}-5& -15 \\ 15 & 5\end{array}\right)=2X+\left(\begin{array}{ll}1& 1 \\ 1 & 1\end{array}\right)\\\\2x=\left(\begin{array}{ll}-6& -16 \\ 14 & 4\end{array}\right)\\\\X=\left(\begin{array}{ll}-3& -8 \\ 7 & 2\end{array}\right)$

Un alt exercitiu cu matrice gasesti aici: https://brainly.ro/tema/9928443

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