\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} \usepackage{amsmath} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Thursday, February 21, 2002 14:53:39} %TCIDATA{LastRevised=Monday, October 03, 2005 09:48:10} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %F=36,\PARA{038
Scientific Notebook: On Line Mathematics\dotfill \thepage} %} \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{Determinanty} \subsection{D\^{o}kaz Crameroveho pravidla} Nech $\mathbf{B}=(b_{1},\ \ldots ,\ b_{n})^{T}$. Matica $\mathbf{A}% =(a_{jk})_{n}^{n}$ je regul\'{a}rna, teda jej hodnos\v{t} a tie\v{z} hodnos% \v{t} matice s\'{u}stavy je $n$, \v{c}o je z\'{a}rove\v{n} po\v{c}et nezn% \'{a}mych. Z toho vypl\'{y}va, \v{z}e s\'{u}stava m\'{a} jedin\'{e} rie\v{s}% enie. Toto rie\v{s}enie m\^{o}\v{z}eme z\'{\i}ska\v{t} aj tak, \v{z}e maticov% \'{u} rovnicu $\mathbf{A}\mathbf{X}=\mathbf{B}$ vyn\'{a}sob\'{\i}me z\v{l}% ava maticou $\mathbf{A}^{-1}$ a dostaneme $\mathbf{X}=\mathbf{A}^{-1}\mathbf{% B}$, pri\v{c}om pre $j$-t\'{u} zlo\v{z}ku plat\'{\i} \begin{equation*} x_{j}=\sum_{k=1}^{n}\left[ (-1)^{j+k}\frac{\det \mathbf{A}_{kj}}{\det \mathbf{A}}\right] b_{k}=\frac{1}{\det \mathbf{A}}% \sum_{k=1}^{n}(-1)^{k+j}b_{k}\det \,\mathbf{A}_{kj}=\frac{\det \mathcal{A}% _{j}}{\det \mathbf{A}}.\;\blacksquare \newline \end{equation*} \begin{center} \begin{tabular}{|c|} \hline \textbf{% %TCIMACRO{\hyperref{Sp\"{a}\v{t}}{}{}{A41.tex#15}}% %BeginExpansion \msihyperref{Sp\"{a}\v{t}}{}{}{A41.tex#15}% %EndExpansion } \\ \hline \end{tabular} \end{center} \end{document}