Question

Matricea: A= 2 1
2 1

Subiectul 2 ex. 1) b. Determinati numarul real x pentru care A*A=xA

Answer 1

$\displaystyle \mathtt{A= \left(\begin{array}{ccc}\mathtt2&\mathtt1\\\mathtt2&\mathtt1\\\end{array}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A \cdot A=xA~~~~~~~~~~~~~~~~~~~~~x=?}$

$\displaystyle \mathtt{A \cdot A= \left(\begin{array}{ccc}\mathtt2&\mathtt1\\\mathtt2&\mathtt1\\\end{array}\right) \cdot \left(\begin{array}{ccc}\mathtt2&\mathtt1\\\mathtt2&\mathtt1\\\end{array}\right) = \left(\begin{array}{ccc}\mathtt{2 \cdot 2+1 \cdot 2}&\mathtt{2 \cdot 1 +1 \cdot 1}\\\mathtt{2 \cdot 2+1 \cdot 2}&\mathtt{2 \cdot 1+1 \cdot1}\\\end{array}\right)=}$

$\displaystyle \mathtt{= \left(\begin{array}{ccc}\mathtt{4+2}&\mathtt{2+1}\\\mathtt{4+2}&\mathtt{2+1}\\\end{array}\right) = \left(\begin{array}{ccc}\mathtt6&\mathtt3\\\mathtt6&\mathtt3\\\end{array}\right)}\\ \\ \mathtt{A \cdot A= \left(\begin{array}{ccc}\mathtt6&\mathtt3\\\mathtt6&\mathtt3\\\end{array}\right) }\\ \\ \mathtt{xA= \left(\begin{array}{ccc}\mathtt{6x}&\mathtt{3x}\\\mathtt{6x}&\mathtt{3x}\\\end{array}\right)}$

$\displaystyle \mathtt{A \cdot A=xA \Rightarrow \left(\begin{array}{ccc}\mathtt6&\mathtt3\\\mathtt6&\mathtt3\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2x}&\mathtt x\\\mathtt{2x}&\mathtt x\\\end{array}\right)} \\ \\ \mathtt{6=2x \Rightarrow x=3;~3=x \Rightarrow x=3}\\ \\ \mathtt{\mathbf{x=3}}$