Question

Va rog sa ma ajutați si pe mine!!!

Answer 1

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