Răspuns :
[tex]A(a)=\left(\begin{array}{ccc}1 & 0 & \ln a \\ 0 & a & 0 \\ 0 & 0 & 1\end{array}\right)[/tex]
a)
Aratati ca det(A(e))=e
Inlocuim pe a cu e si adaugam primele doua linii ale determinantului
[tex]det(A(e))=\left|\begin{array}{ccc}1 & 0 & \ln e \\ 0 & e & 0 \\ 0 & 0 & 1\end{array}\right|=\left|\begin{array}{ccc}1 & 0 &1 \1 \\ 0 & e & 0 \\ 0 & 0 & 1\end{array}\right|[/tex]
1 0 1
0 e 0
det(A(e))=(e+0+0)-(0+0+0)=e
b)
[tex]det(A(a^2))=det(A(a)A(a))[/tex]
Calculam A(a)A(a), iar apoi ii calculam determinantul
[tex]A(a)\cdot A(a)=\left(\begin{array}{ccc}1 & 0 & \ln a \\ 0 & a & 0 \\ 0 & 0 & 1\end{array}\right)\cdot \left(\begin{array}{ccc}1 & 0 & \ln a \\ 0 & a & 0 \\ 0 & 0 & 1\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 2\ln a \\ 0 & a^2 & 0 \\ 0 & 0 & 1\end{array}\right)[/tex]
[tex]\left|\begin{array}{ccc}1 & 0 & 2\ln a \\ 0 & a^2 & 0 \\ 0 & 0 & 1\end{array}\right|[/tex]
1 0 2lna
0 a² 0
=(a²+0+0)-(0+0+0)=a²
[tex]det(A(a^2))=\left|\begin{array}{ccc}1 & 0 & \ln a^2 \\ 0 & a^2 & 0 \\ 0 & 0 & 1\end{array}\right|=\left|\begin{array}{ccc}1 & 0 & 2\ln a \\ 0 & a^2 & 0 \\ 0 & 0 & 1\end{array}\right|[/tex]
1 0 2lna
0 a² 0
=(a²+0+0)-(0+0+0)=a²
Observam ca sunt egali cei doi determinanti calculati mai sus
c)
A(a)+A(b)=2A(a)A(b)
Ne folosim de punctul b
A(a)A(b)=A(ab)
[tex]A(ab)=\left(\begin{array}{ccc}1 & 0 & \ln ab \\ 0 & ab & 0 \\ 0 & 0 & 1\end{array}\right)[/tex]
[tex]A(a)+A(b)=\left(\begin{array}{ccc}1 & 0 & \ln a \\ 0 & a & 0 \\ 0 & 0 & 1\end{array}\right)+\left(\begin{array}{ccc}1 & 0 & \ln b \\ 0 & b & 0 \\ 0 & 0 & 1\end{array}\right)=\left(\begin{array}{ccc}2 & 0 & \ln ab \\ 0 & a+b & 0 \\ 0 & 0 & 2\end{array}\right)[/tex]
2A(a)A(b)=2A(ab)
A(a)+A(b)=2A(ab)
[tex]\left(\begin{array}{ccc}2 & 0 & \ln ab \\ 0 & a+b & 0 \\ 0 & 0 & 2\end{array}\right)=\left(\begin{array}{ccc}2 & 0 & 2\ln ab \\ 0 & 2ab & 0 \\ 0 & 0 & 2\end{array}\right)[/tex]
Egalam termenii si obtinem:
a+b=2ab
ln(ab)=2ln(ab)
ln(ab)=0
ab=1
a+b=2ab
ab=1
a+b=2
ab=1
S=2 si P=1
t²-St+P=0
t²-2t+1=0
(t-1)²=0
t=1⇒ a=1 si b=1
Un alt exercitiu cu matrice gasesti aici: https://brainly.ro/tema/9685683
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