Buna ziua, tocmai am inceput si eu studiul despre matrice. Daca se poate sa verificati un exercitiu rezolvat de mine 🙂

Fie matricele:

a). Sa se arate ca A(x)A(y)=A(x+y) si B(x)B(y)=B(xy).
b). Folosind aceste rezultate, sa se gaseasca puterea n a matricei A(x),respectiv B(x), pt n>=1.

*** QuickLaTeX cannot compile formula:

	
	\[
	\begin{array}{l}
	 a).\;A(x)A(y) = \left( {\begin{array}
	   1 & 0 & x  \\
	   { - x} & 1 & { - \frac{1}{2}x^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right) \cdot \left( {\begin{array}
	   1 & 0 & y  \\
	   { - y} & 1 & { - \frac{1}{2}y^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right) = \left( {\begin{array}
	   1 & 0 & {x + y}  \\
	   { - \left( {x + y} \right)} & 1 & { - \frac{1}{2}\left( {x + y} \right)^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right) = A(x + y) \\
	  \\
	 B(x)B(y) = \left( {\begin{array}
	   1 & {\ln x} & 0  \\
	   0 & 1 & 0  \\
	   0 & 0 & x  \\
	\end{array}} \right) \cdot \left( {\begin{array}
	   1 & {\ln y} & 0  \\
	   0 & 1 & 0  \\
	   0 & 0 & y  \\
	\end{array}} \right) = \left( {\begin{array}
	   1 & {\ln xy} & 0  \\
	   0 & 1 & 0  \\
	   0 & 0 & {xy}  \\
	\end{array}} \right) = B(xy) \\
	  \\
	 b).\;A^n (x) = ? \\
	 Calculand\;A^2 (x) = A(x) \cdot A(x) = \left( {\begin{array}
	   1 & 0 & {2x}  \\
	   { - 2x} & 1 & { - \frac{4}{2}x^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right)\;;\quad A^3 (x) = \left( {\begin{array}
	   1 & 0 & {3x}  \\
	   { - 3x} & 1 & { - \frac{9}{2}x^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right)\;\quad ;\;A^4 (x) = \left( {\begin{array}
	   1 & 0 & {4x}  \\
	   { - 4x} & 1 & { - \frac{{16}}{2}x^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right) \\
	 Se\;observa\;ca\;A^n (x) = \left( {\begin{array}
	   1 & 0 & {nx}  \\
	   { - nx} & 1 & { - \frac{1}{2}n^2 x^2 }  \\
	   0 & 0 & 1  \\
	\end{array}} \right) \\
	 \Pr in\;inductie\;se\;demonstreaza\;ca\;P(n)\;este\;adev. \\
	 P(n) \to P(n + 1) \\
	 P(n)]

*** Error message:
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*** QuickLaTeX cannot compile formula:

	\[
	\begin{array}{l}
	 Ana\log \;,\;calculand\;B^2 (x);B^3 (x);B^4 (x)\;se\;deduce\;ca]
	

*** Error message:
\begin{array} on input line 10 ended by \end{document}.
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\begin{array} on input line 10 ended by \end{document}.
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Emergency stop.

.

Este bine? 😀