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%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.00.0.2570}
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%TCIDATA{Created=Thursday, February 21, 2002 17:34:45}
%TCIDATA{LastRevised=Monday, October 03, 2005 09:43:44}
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%TCIDATA{}
%TCIDATA{CSTFile=On line bluem.cst}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}
\section{Matice a s\'{u}stavy line\'{a}rnych rovn\'{\i}c}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{aindex.tex}}%
%BeginExpansion
\msihyperref{Obsah}{}{}{aindex.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Obsah kapitoly}{}{}{A2.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{A2.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{OA2.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{OA2.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Index}{}{}{INDA.tex}}%
%BeginExpansion
\msihyperref{Index}{}{}{INDA.tex}%
%EndExpansion
} \\ \hline
\end{tabular}
\end{center}
\subsection{Cvi\v{c}enia}
\begin{enumerate}
\item Zistite, ktor\'{e} z mat\'{\i}c
\[
\begin{array}{rcl}
\mathbf{A}=\left(
\begin{array}{rr}
1 & 0 \\
2 & 1 \\
-1 & 2%
\end{array}%
\right) , & \mathbf{B}=(-1,\;0,\;2), & \mathbf{D}=\left(
\begin{array}{rrrr}
1 & 0 & 0 & -1 \\
0 & 2 & 0 & 5%
\end{array}%
\right) , \\
\mathbf{E}=\left(
\begin{array}{rr}
1 & 1 \\
1 & 2%
\end{array}%
\right) , & \mathbf{F}=\left(
\begin{array}{r}
1 \\
0 \\
3 \\
4%
\end{array}%
\right) , & \mathbf{G}=\left(
\begin{array}{rrr}
1 & 2 & 0 \\
-2 & 0 & -3 \\
0 & 3 & 5%
\end{array}%
\right) , \\
\mathbf{H}=\left(
\begin{array}{rrrr}
0 & 1 & 0 & 3 \\
0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0%
\end{array}%
\right) , & \mathbf{J}=\left(
\begin{array}{rr}
1 & 0 \\
0 & -7%
\end{array}%
\right) , & \mathbf{K}=\left(
\begin{array}{rr}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha%
\end{array}%
\right) , \\
\mathbf{L}=\left(
\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0, & 1, & 2, & 0 \\
0, & 0, & 0, & 1%
\end{array}%
\right) , & \mathbf{M}=\left(
\begin{array}{rrr}
2, & 1, & 2 \\
0, & 0, & 2 \\
0, & 0, & -2 \\
0, & 0, & 0%
\end{array}%
\right) , & \mathbf{O}=\left(
\begin{array}{rrrrr}
1, & 0, & 0, & 1, & 0 \\
0, & 0, & 1, & 5, & 0 \\
0, & 0, & 0, & ,0, & 1%
\end{array}%
\right)%
\end{array}%
\]%
s\'{u}
\begin{enumerate}
\item stup\v{n}ovit\'{e},\qquad
\CustomNote{Answer}{$\mathbf{B},\,\mathbf{D},\,\mathbf{H},\,\mathbf{J},\,%
\mathbf{K}$ pre $\alpha =k\pi ,\,k\in \mathbf{Z},\,\mathbf{L},\,\mathbf{O}$}
\item redukovan\'{e} stup\v{n}ovit\'{e}?\qquad
\CustomNote{Answer}{$\mathbf{K}$ pre $\alpha =2k\pi ,\,k\in \mathbf{Z},\,%
\mathbf{L},\,\mathbf{O}$}
\end{enumerate}
\item N\'{a}jdite redukovan\'{u} stup\v{n}ovit\'{u} maticu, ktor\'{a} je
riadkovo ekvivaletn\'{a} s maticou
\begin{enumerate}
\item $\left(
\begin{array}{rrrr}
1 & -2 & 0 & 1 \\
3 & 2 & 1 & 0 \\
1 & -1 & 1 & -2%
\end{array}%
\right) \qquad
\CustomNote{Answer}{$\left(
\begin{array}{rrrr}
1, & 0, & 0, & 1 \\
0, & 1, & 0, & 0 \\
0, & 0, & 1, & -3%
\end{array}%
\right) $}$\medskip
\item $\left(
\begin{array}{rrrrrr}
4 & 5 & 6 & 7 & 8 & 9 \\
3 & 4 & 5 & 6 & 7 & 8 \\
2 & 3 & 4 & 5 & 6 & 7 \\
1 & 2 & 3 & 4 & 5 & 6%
\end{array}%
\right) \qquad
\CustomNote{Answer}{$\left(
\begin{array}{rrrrrr}
1, & 0, & -1, & -2, & -3, & -4 \\
0, & 1, & 2, & 3, & 4, & 5 \\
0, & 0, & 0, & 0, & 0, & 0 \\
0, & 0, & 0, & 0, & 0, & 0%
\end{array}%
\right) $}$
\end{enumerate}
\item Rozhodnite, \v{c}i s\'{u} ekvivaletn\'{e} s\'{u}stavy line\'{a}rnych
rovn\'{\i}c $\mathcal{S}_{1}\mathcal{S}_{2}$, ak
\begin{enumerate}
\item $\mathcal{S}_{1}:%
\begin{array}{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2x_{1} & + & x_{2} & - & 3x_{3} & = & 2 \\
x_{1} & - & 3x_{2} & - & 2x_{3} & = & 1%
\end{array}%
$ \qquad $\mathcal{S}_{2}:%
\begin{array}{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2y_{1} & + & y_{2} & - & 3y_{3} & = & 2 \\
y_{1} & + & 4y_{2} & - & y_{3} & = & 1%
\end{array}%
\qquad
\CustomNote{Answer}{\'{a}no}$
\item $\mathcal{S}_{1}:%
\begin{array}{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2x_{1} & + & x_{2} & - & 3x_{3} & = & 2 \\
x_{1} & - & 3x_{2} & - & 2x_{3} & = & 1%
\end{array}%
$ \qquad $\mathcal{S}_{2}:%
\begin{array}{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2x_{1} & + & x_{2} & - & 3x_{3} & = & 2 \\
3x_{1} & - & 2x_{2} & - & x_{3} & = & 3%
\end{array}%
\qquad
\CustomNote{Answer}{%
nie}$
\end{enumerate}
\item Rie\v{s}te s\'{u}stavu line\'{a}rnych rovn\'{\i}c\vspace{0pt}
\begin{enumerate}
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
3x_{1} & - & 2x_{2} & + & 3x_{3} & = & 3 \\
-2x_{1} & + & 3x_{2} & - & 5x_{3} & = & -3 \\
x_{1} & + & 7x_{2} & - & 3x_{3} & = & 16%
\end{array}%
\qquad
\CustomNote{Answer}{$K=\{(1,\,3,\,2)\}$}$
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}rr@{\;=\;}r}
-2x_{1} & + & 3x_{2} & + & 5x_{3} & = & 3 \\
3x_{1} & + & x_{2} & - & 4x_{3} & = & 5 \\
4x_{1} & + & 5x_{2} & - & 3x_{3} & = & 8%
\end{array}%
\qquad
\CustomNote{Answer}{%
nem\'{a} rie\v{s}enie}$
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}r@{}r@{}rr@{\;=\;}r}
5x_{1} & + & 2x_{2} & + & 4x_{3} & + & x_{4} & = & 9 \\
-2x_{1} & + & x_{2} & - & x_{3} & - & x_{4} & = & 5 \\
-4x_{1} & + & 4x_{2} & & & - & 2x_{4} & = & 18 \\
7x_{1} & + & 3x_{2} & + & 7x_{3} & + & 2x_{4} & = & 12 \\
3x_{1} & + & 2x_{2} & + & 2x_{3} & & & = & 10%
\end{array}%
\qquad
\CustomNote{Answer}{$K=\left\{ \left( \frac{2}{3},\,\frac{31}{6}+t,\,-\frac{7%
}{6}-t,\,2t\right) ,\;t\in \mathbf{R}\right\} $}$
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2x_{1} & - & 3x_{2} & + & x_{3} & - & x_{4} & = & 3 \\
-x_{1} & - & 2x_{2} & - & x_{3} & + & 2x_{4} & = & -2 \\
-x_{1} & - & 9x_{2} & - & 2x_{3} & + & 5x_{4} & = & 0%
\end{array}%
\qquad
\CustomNote{Answer}{%
nem\'{a} rie\v{s}enie}$
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}rr@{\;=\;}r}
2x_{1} & - & 2x_{2} & - & 3x_{3} & + & x_{4} & - & 4x_{5} & = & 11 \\
3x_{1} & - & 3x_{2} & + & 2x_{3} & - & 2x_{4} & + & x_{5} & = & -7 \\
5x_{1} & - & 5x_{2} & - & 2x_{3} & + & 3x_{4} & - & 11x_{5} & = & 18 \\
2x_{1} & - & 2x_{2} & + & x_{3} & & & - & 2x_{5} & = & 0%
\end{array}%
\qquad
\CustomNote{Answer}{$K=\left\{ \left( 1+a+b,\,a,\,-2,\,3+2b,\,b\right)
,\;a,b\in \mathbf{R}\right\} $}$
\item $%
\begin{array}[t]{r@{}r@{}r@{}r@{}r@{}r@{}r@{}r@{}rr@{\;=\;}r}
6x_{1} & - & 2x_{2} & + & 2x_{3} & + & 5x_{4} & + & 7x_{5} & = & 0 \\
9x_{1} & - & 3x_{2} & + & 4x_{3} & + & 8x_{4} & + & 9x_{5} & = & 0 \\
-3x_{1} & + & x_{2} & - & 6x_{3} & - & 5x_{4} & + & 4x_{5} & = & 0 \\
3x_{1} & - & x_{2} & + & 4x_{3} & + & 4x_{4} & - & x_{5} & = & 0%
\end{array}%
\qquad
\CustomNote{Answer}{$K=\left\{ \left( \dfrac{a-2b-5c}{3},\,a,\,\dfrac{-b+3c}{%
2},\,b,\,c\right) ,\;a,b,c\in \mathbf{R}\right\} $}$
\end{enumerate}
\end{enumerate}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{aindex.tex}}%
%BeginExpansion
\msihyperref{Obsah}{}{}{aindex.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Obsah kapitoly}{}{}{A2.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{A2.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{OA2.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{OA2.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Index}{}{}{INDA.tex}}%
%BeginExpansion
\msihyperref{Index}{}{}{INDA.tex}%
%EndExpansion
} \\ \hline
\end{tabular}
\end{center}
\rule{6.5in}{0.04in}
\textsl{Matematika 1}
\section{Matice a s\'{u}stavy line\'{a}rnych rovn\'{\i}c}
\end{document}