Question

ma poate ajuta cineva . nu inteleg ?

Answer 1

Se observa ca $A^{2} -4A+4I_{2} =O_{2}$ mai poate fi scris ca $A^{2}-2*A*I_{2}-2*A*I_{2}+2^{2}I_{2}^{2}=O_{2}$. Scotand in factor obtinem $A(A-I_{2})-I_{2}(A-I_{2})=(A-I_{2})*(A-I_{2})=(A-I_{2})^{2}=O_{2}$
Dam o forma generala matricei A, astfel $A=\left[\begin{array}{ccc}x&y\\y&x\end{array}\right]$. Neavand cerinta originala, presupun ca e vorba de astfel de matrice.

Revenind la $(A-I_{2})^{2}=O_{2}$ si inlocuind pe A cu forma data, se obtine 

$(\left[\begin{array}{ccc}x&y\\y&x\end{array}\right]-\left[\begin{array}{ccc}2&0\\0&2\end{array}\right])^{2}=\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$

$(\left[\begin{array}{ccc}x-2&y\\y&x-2\end{array}\right])^{2}=\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$