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z=1Y45%KHKw*!um~^Hix+<3`~73H52Jvi?fO`t_ew?b$h9R;-7W7WAF%sZ)wTSWDd0y0>NMw=7Q!HP>=u@h>wpC1{H$x@$6uC z-n279Iay44bu{CCZ%U`=xV(Z*Ay`5pYO0&T{wGKF2D0%_eB*T%$6fM=Ylk}|6{QIS?(^8Te_%U*%G?NV5x?*~0@e!`qtXVh=bWFE@p z+b3Eg=fh%}+bqd&JI?UJQL>zCmDOW$s;t|V6WIc9POF>yjdO*^_iN-zcN0{0NEB!P derT?4(06X0?<`^1g4{w}Lf8xpGODuJ{{YpBN;m)j diff --git a/linear-algebra/exercise/1-determinant/determinant.tex b/linear-algebra/exercise/1-determinant/determinant.tex index df20538..4d3e606 100644 --- a/linear-algebra/exercise/1-determinant/determinant.tex +++ b/linear-algebra/exercise/1-determinant/determinant.tex @@ -22,6 +22,8 @@ % 数学公式 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 +\usepackage{rotating} +% 用于旋转对象(旋转包) \author{Didnelpsun} \title{行列式} \date{} @@ -76,16 +78,440 @@ $\therefore\,-a_{11}a_{23}a_{32}a_{44}+a_{11}a_{23}a_{34}a_{42}$。 -\section{证明行列式值} - -与计算行列式值的题型不同的是,其行列式的值是固定给出的,一方面虽然约束了解题思路,一方面也给出了解题的方向,需要结果与给定值“靠近”。 - -\textbf{例题:} - -\section{计算行列式值} +\section{行列式} 包含直接计算行列式的值和已知行列式值计算参数值两种体型,基本上求解方式一致。 +证明行列式值与计算行列式值的题型不同的是,其行列式的值是固定给出的,一方面虽然约束了解题思路,一方面也给出了解题的方向,需要结果与给定值“靠近”。 + +\subsection{基本行列式与计算} + +\subsubsection{三角行列式} + +$\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + & \ddots & \cdots & a_{2n} \\ + & & \ddots & \vdots \\ + & & & a_{nn} +\end{array}\right|= +\left|\begin{array}{cccc} + a_{11} & & & \\ + a_{21} & \ddots & & \\ + \vdots & \cdots & \ddots & \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right|= +\left|\begin{array}{cccc} + a_{11} & & & \\ + & \ddots & & \\ + & & \ddots & \\ + & & & a_{nn} +\end{array}\right|=a_{11}\cdots a_{nn}$ + +\subsubsection{反三角行列式} + +$\left|\begin{array}{cccc} + & & & a_{1n} \\ + & & \begin{turn}{80}$\ddots$\end{turn} & a_{2n} \\ + & \begin{turn}{80}$\ddots$\end{turn} & \cdots & \vdots \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right|= +\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & \cdots & \begin{turn}{80}$\ddots$\end{turn} & \\ + \vdots & \begin{turn}{80}$\ddots$\end{turn} & & \\ + a_{n1} & & & +\end{array}\right|= +\left|\begin{array}{cccc} + & & & a_{1n} \\ + & & \begin{turn}{80}$\ddots$\end{turn} & \\ + & \begin{turn}{80}$\ddots$\end{turn} & & \\ + a_{n1} & & & +\end{array}\right|=(-1)^{\frac{n(n-1)}{2}}a_{1n}\cdots a_{n1}$ + +\subsubsection{范德蒙德行列式} + +$\left|\begin{array}{cccc} + 1 & 1 & \cdots & 1 \\ + a_1 & a_2 & \cdots & a_n \\ + \cdots & \cdots & \vdots & \cdots \\ + a_1^{n-1} & a_2^{n-1} & \cdots & a_n^{n-1} \\ +\end{array}\right|=\sum\limits_{1\leqslant j