[tex] Nicio~problema,~o~rezolv~eu!\\ \\ A=$\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{pmatrix} $~si~ I_3=$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $\\ \\ A*A=$\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{pmatrix} $*$\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{pmatrix} $=$\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{pmatrix} $.\\ \\ A^2*A= $\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{pmatrix} * $\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 1 \\ 1 & -1 & 0 \end{pmatrix} = $\begin{pmatrix} 3 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 2 \end{pmatrix} \\ \\ x*A= $\begin{pmatrix} x & 0 & x \\ 0 & -x & x \\ x & -x & 0 \end{pmatrix} ~si~y*I_3=$\begin{pmatrix} y & 0 & 0\\ 0 & y & 0 \\ 0 & 0 & y \end{pmatrix} \\ \\ Daca~le~aduni,~vei~avea:\\ \\ xA+yI_3= $\begin{pmatrix} x+y & 0 & x \\ 0 & y-x & x \\ x & -x & y \end{pmatrix} \\ \\ \\ Egaland~A^3~cu~suma~asa:\\ \\ $\begin{pmatrix} 3 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 2 \end{pmatrix} =$\begin{pmatrix} x+y & 0 & x \\ 0 & y-x & x \\ x & -x & y \end{pmatrix}~\Leftrightarrow\left \{ {{x=1} \atop {y=2}} } \right. [/tex]
