[tex]\bf {1.}\\ \\ \it b)\ \ x\cdot A= x\cdot\begin{pmatrix} \it 3 & \it 6\\\it 1 & \it 2\end{pmatrix} = \begin{pmatrix}\it 3x & 6x\\x & 2x\end{pmatrix}\\ \\ \\ A\cdot A= \begin{pmatrix}\it 3 & \it 6\\\it 1 & \it 2\end{pmatrix}\cdot \begin{pmatrix}\it 3 & \it 6\\\it 1 & \it 2\end{pmatrix}=\begin{pmatrix}\it 15 & \it 30\\\it 5 & \it 10\end{pmatrix}[/tex]
[tex]\it x\cdot A=A\cdot A \Rightarrow\ \begin{pmatrix} \it 3x & \it 6x\\\it x & \it 2x\end{pmatrix} = \begin{pmatrix} \it 15 & \it 30\\\it 5 & \it 10\end{pmatrix} \Rightarrow x=5[/tex]
[tex]\it c)\ A+aI_2=\begin{pmatrix} \it 3 & \it 6\\\it 1 & \it 2\end{pmatrix} +\begin{pmatrix} \it a & \it 0\\\it 0 & \it a\end{pmatrix} =\begin{pmatrix} \it 3+a & \it 6\\\it 1 & \it 2+a\end{pmatrix}\\ \\ \\ det(A+aI_2)=0 \Rightarrow \begin{vmatrix} \it 3+a & \it 6\\\it 1 & \it 2+a\end{vmatrix} = 0 \Rightarrow (3+a)(2+a)-6=0 \Rightarrow\\ \\ \\ \Rightarrow a^2+5a+6-6=0 \Rightarrow a^2+5a=0 \Rightarrow a(a+5)=0 \Rightarrow a_1=-5,\ \ a_2=0[/tex]
