Question

va rog ajutati-ma cu exercitiile astea 2 si 3 :)

Answer 1

[tex]A^2-4A+4I_2=O_2\Leftrightarrow(A-2I_2)^2=O_2\Leftrightarrow \left[\begin{array}{ccc}x-2&y\\y&x-2\end{array}\right]^2=O_2\\ \Leftrightarrow \left[\begin{array}{ccc}(x-2)^2+y^2&2y(x-2)\\2y(x-2)&(x-2)^2+y^2\end{array}\right]=O_2 \\ \Leftrightarrow(x-2)^2+y^2=0,2y(x-2)=0\Leftrightarrow x=2,y=0 [/tex]

[tex] A=\left[\begin{array}{ccc}2&0\\0&2\end{array}\right] \\ \text{Se demonstreaza prin inductie ca }\\ A^2=\left[\begin{array}{ccc}2^n&0\\0&2^n\end{array}\right] [/tex]

$\text{La problema 3 se efectueaza caculele i se obtine \alpha=3,\beta=-3 }$