Răspuns :
[tex]A(x)=\left(\begin{array}{cc}2^{x} & 0 \\ 0 & 3^{x}\end{array}\right)[/tex]
a)
Calculam det(A(x)), facem diferenta dintre produsul diagonalelor
det(A(x))=2ˣ × 3ˣ-0=6ˣ
b)
[tex]\left(\begin{array}{cc}2^{x} & 0 \\ 0 & 3^{x}\end{array}\right)\cdot \left(\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right)=\left(\begin{array}{cc}2^{x} & 2^x \\ 0 & 3^{x}\end{array}\right)[/tex]
[tex]\left(\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right)\cdot \left(\begin{array}{cc}2^{x} & 0 \\ 0 & 3^{x}\end{array}\right)=\left(\begin{array}{cc}2^{x} & 3^x \\ 0 & 3^{x}\end{array}\right)[/tex]
[tex]2^x=3^x\\\\x=0[/tex]
c)
[tex]Fie\ X=\left(\begin{array}{cc}a& b\\ c & d\end{array}\right)[/tex]
[tex]X\cdot X=\left(\begin{array}{cc}a& b\\ c & d\end{array}\right)\cdot \left(\begin{array}{cc}a& b\\ c & d\end{array}\right)=\left(\begin{array}{cc}a^2+bc&ab+bd\\ ac+cd& bc+d^2\end{array}\right)\\\\\left(\begin{array}{cc}a^2+bc&ab+bd\\ ac+cd& bc+d^2\end{array}\right)=\left(\begin{array}{cc}2&0\\ 0&3\end{array}\right)[/tex]
a²+bc=2
b(a+d)=0
b=0 si a+d=0
a=-d
c(a+d)=0
c=0
a=-d
bc+d²=3
d²=3⇒ d=±√3 care este numar irational
a²+bc=2
a²=2
a=±√2 care este numar irational
Un alt exercitiu cu matrice gasesti aici: https://brainly.ro/tema/9919039
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