Question

Subiectul II (a,b,c)
E urgent!!! Dau coroniță!!

Answer 1

$\it a)\ \ detM = \begin{vmatrix}2&1\\1&2 \end{vmatrix} =2\cdot2-1\cdot1=4-1=3\\ \\ \\ b)\ A(a)\cdot A(a)=\begin{pmatrix}a&1\\3&2 \end{pmatrix}\cdot\begin{pmatrix}a&1\\3&2 \end{pmatrix} =\begin{pmatrix}a^2+3&a+2\\3a+6&7 \end{pmatrix}\ \ \ \ \ (1)\\ \\ \\ 4A(a)-I_2=\begin{pmatrix}\it4a&\it4\\\it12&\it8 \end{pmatrix}-\begin{pmatrix}1&0\\0&1 \end{pmatrix}=\begin{pmatrix}4a-1&4\\12&7 \end{pmatrix}\ \ \ \ \ (2)\\ \\ \\ A(a)\cdot A(a)=4A(a)-I_2 \ \ \ \ \ (3)$

$\it (1),\ (2),\ (3)\Rightarrow\ \begin{cases}\it a^2+3=4a-1\Rightarrow a^2-4a+4=0 \Rightarrow (a-2)^2=0\Rightarrow a=2\\ \\ \it a+2=4 \Rightarrow a=2\\ \\ \it3a+6=12 \Rightarrow 3a=6\Rightarrow a=2\end{cases}$