京都府立医科大学 前期 2000年度 問3

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大学名 京都府立医科大学
学科・方式 前期
年度 2000年度
問No 問3
学部 医学部
カテゴリ 行列と連立一次方程式
状態 解答なし 解説なし ウォッチリスト

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% \newcommand{\枠}[1]{\fboxsep1pt\fbox{ #1 }\,} %=========================================================== %=========================================================== % ■≦と≧の定義 %----------------------------------------------------------- \def\le{\leqq} \def\ge{\geqq} \newcommand{\LEQQ}{\leqq} \newcommand{\GEQQ}{\geqq} % \newcommand{\LEQQ}{\mathrel{\mathpalette\gl@align<}} % \newcommand{\GEQQ}{\mathrel{\mathpalette\gl@align>}} % \newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@ % \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}} % エラーになる。理由はよく解らん。 %=========================================================== %=========================================================== % ■行列 2×2 %----------------------------------------------------------- \newcommand{\matrixTT}[4]{ \begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix} } %=========================================================== %=========================================================== % ■行列 3×3 %----------------------------------------------------------- \newcommand{\matrixTTT}[9]{ \begin{pmatrix} #1 & #2 & #3\\ #4 & #5 & #6\\ #7 & #8 & #9 \end{pmatrix} } %=========================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ■ 3*1行列の定義 \newcommand{\matrixthreeone}[3]{ \left(\begin{array}{c} #1\\ #2\\ #3 \end{array}\right) } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %=========================================================== % ■角括弧 複数行 #1 は横幅%,#2は内容 % %----------------------------------------------------------- \newcommand{\角括弧}[2]{% $% \left[ \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right]% $% } %=========================================================== %=========================================================== % ■丸括弧 複数行 #1 は横幅%,#2は内容 %----------------------------------------------------------- \newcommand{\丸括弧}[2]{% $% \left( \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right)% $% } %=========================================================== %=========================================================== % ■丸囲み数字 ①とか %----------------------------------------------------------- \newcommand{\tyoMaru}[1]{\mbox{{\normalsize \textcircled{\scriptsize #1}}}} %=========================================================== %=========================================================== % ■ 微分 df/dt %----------------------------------------------------------- \newcommand{\dd}[2]{% \dfrac{{\rm d}#1}{{\rm d}#2}% } %=========================================================== %=========================================================== % ■ ∫記号のdisplaystyleマクロ %----------------------------------------------------------- % \newcommand{\dint}{\displaystyle\int}% emathと干渉しないように\defで定義 \def\dint{\displaystyle\int} %=========================================================== %=========================================================== % ■ Σ記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dsum}{\displaystyle\sum} %=========================================================== %=========================================================== % ■ lim記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dlim}{\displaystyle\lim} %=========================================================== %=========================================================== % ■ nCr マクロ %----------------------------------------------------------- \newcommand{\nCr}[2]{% {}_{#1}\mathrm{C}_{#2}% } %=========================================================== %=========================================================== % ■ 証明終了記号 ■ %----------------------------------------------------------- \newcommand{\■}{{\tiny \text{■}}} %=========================================================== \begin{document} %%%%% ■ 本文開始 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{FRAME}% この間に設問を書く \noindent 実数を成分にもつ3次の正方行列 $A=% \matrixTTT% {a_{11}}{a_{12}}{a_{13}}% {a_{21}}{a_{22}}{a_{23}}% {a_{31}}{a_{32}}{a_{33}}$% は次を満たすとする. \begin{enumerate} \item[\mbox{[1]}] すべての$i$,$j$について$a_{ij}=a_{ji}$である. \item[\mbox{[2]}] $A^2=A$ \end{enumerate} このとき,次を示せ. \begin{enumerate} \item 各$i\ (i=1,\,2,\,3)$に対して,$0\LEQQ a_{ij}\LEQQ 1$が成り立つ. \item $i\neq j$のとき,$|a_{ii}|\LEQQ 1/2$が成り立つ. \item $a_{ii}=0$または$a_{ii}=1$のとき,各$j\ (j\neq i)$について$a_{ij}=0$が成り立つ. \item 行列$A$が逆行列をもつならば,$A$は単位行列である. \end{enumerate} さらに, \begin{enumerate} \setcounter{enumi}{4} \item $a_{11}=1$,$a_{23}\neq 0$であるような行列$A$の例をあげよ. \end{enumerate} % \end{FRAME} %--- 解答 ------------------------------------------------------------------------ %%%%%%% ■ 本文終了 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}