Răspuns :
[tex]A=\left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)[/tex]
1)
Calculam detA, facem diferenta dintre produsul diagonalelor
detA=3-9=-6
2)
[tex]A\cdot B=\left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)\cdot \left(\begin{array}{ll}-\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{6} \end{array}\right)=\left(\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right)=I_2[/tex]
3)
[tex]A\cdot A=\left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)\cdot \left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)=\left(\begin{array}{ll}10 & 12 \\ 12 & 18\end{array}\right)\\\\\left(\begin{array}{ll}10 & 12 \\ 12 & 18\end{array}\right)-\left(\begin{array}{ll}4& 12 \\ 12 & 12\end{array}\right)=\left(\begin{array}{ll}6 &0 \\ 0&6\end{array}\right)=6I_2[/tex]
4)
det(A+xI₂)=-1
(1-x)(3-x)-9=-1
3-4x+x²-9+1=0
x²-4x-5=0
Δ=16+20=36
[tex]x_1=\frac{4-6}{2} =-1\\\\x_2=\frac{4+6}{2} =5[/tex]
5)
Din punctul 3 avem :
A·A-4A=6I₂
A·A=4A+6I₂
A·A·A=(4A+6I₂)·A=4A·A+6I₂·A=4A·A+6A=4(4A+6I₂)+6A=16A+24I₂+6A=22A+24I₂
22A+24I₂=aA+24I₂
22A=aA
a=22
6)
[tex]A\cdot X=\left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)\cdot \left(\begin{array}{ll}2& 1 \\ a & b\end{array}\right)=\left(\begin{array}{ll}2+3a & 1+3b \\ 6+3a & 3+3b\end{array}\right)[/tex]
[tex]X\cdot A=\left(\begin{array}{ll}2& 1 \\ a & b\end{array}\right)\cdot \left(\begin{array}{ll}1 & 3 \\ 3 & 3\end{array}\right)=\left(\begin{array}{ll}5 & 9 \\ a+3b & 3a+3b\end{array}\right)[/tex]
2+3a=5
3a=3
a=1
1+3b=9
3b=8
[tex]b=\frac{8}{3}[/tex]
Un alt exercitiu cu matrice gasesti aici: https://brainly.ro/tema/9919034
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