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<TITLE> 利用矩陣解法 </TITLE>
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<H1>6.1 利用矩陣解法</H1>
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假設(shè)一組聯(lián)立線性方程式為
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我們習(xí)慣將上組方程式以矩陣方式表示如下
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<I> </I><FONT FACE="Times New Roman">AX=B</FONT>
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其中<FONT FACE="Times New Roman"> A </FONT>為等式左邊各方程式的系數(shù)項，<FONT FACE="Times New Roman">X
</FONT>為欲求解的未知項，<FONT FACE="Times New Roman">B </FONT>代表等式右邊之已知項
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<A NAME="work"> <IMG SRC="img00002-3.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img6/img00002.gif"></A>
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要解上述的聯(lián)立方程式，我們可以利用在第六章介紹的矩陣左除<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000>\</FONT><FONT FACE="Times New Roman">
</FONT>做運(yùn)算，即是<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=A\B</FONT>。
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如果將原方程式改寫成<FONT FACE="Times New Roman"> XA=B</FONT>，且令<FONT FACE="Times New Roman">
X, A </FONT>和<FONT FACE="Times New Roman"> B </FONT>分別為
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注意上式的<FONT FACE="Times New Roman"> X, B </FONT>已改寫成列向量，<FONT FACE="Times New Roman">A</FONT>其實(shí)是前一個方程式中<FONT FACE="Times New Roman">
A </FONT>的轉(zhuǎn)置矩陣。上式的<FONT FACE="Times New Roman"> X </FONT>可以矩陣右除<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">/</FONT><FONT FACE="Times New Roman">
</FONT>求解，即是<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=B/A</FONT>。
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若以反矩陣運(yùn)算求解<FONT FACE="Times New Roman"> AX=B, X=<IMG SRC="img00004-3.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img6/img00004.gif">B</FONT>，即是<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=inv(A)*B</FONT>，或是改寫成<FONT FACE="Times New Roman">
XA=B, X=B<IMG SRC="img00005-3.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img6/img00005.gif"></FONT>，即是<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=B*inv(A)</FONT>。
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我們直接以下面的例子來說明這三個運(yùn)算的用法：
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A=[3 2 -1;
-1 3 2; 1 -1 -1]; % </FONT><FONT COLOR=#FF0000>將等式的左邊系數(shù)鍵入</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> B=[10 5 -1]';
% </FONT><FONT COLOR=#FF0000>將等式右邊之已知項鍵入，</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">B</FONT><FONT COLOR=#FF0000>要做轉(zhuǎn)置</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> X=A\B
% </FONT><FONT COLOR=#FF0000>先以左除運(yùn)算求解</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
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<FONT COLOR=#FF0000 FACE="Times New Roman">X = % </FONT><FONT COLOR=#FF0000>注意</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X</FONT><FONT COLOR=#FF0000>為行向量</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -2</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 5</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 6</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> C=A*X
% </FONT><FONT COLOR=#FF0000>驗(yàn)算解是否正確</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">C = % C=B</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 10</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 5</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -1<BR>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A=A';
% </FONT><FONT COLOR=#FF0000>將</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT><FONT COLOR=#FF0000>先做轉(zhuǎn)置</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> B=[10 5 -1];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> X=B/A
% </FONT><FONT COLOR=#FF0000>以右除運(yùn)算求解的結(jié)果亦同</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">X = % </FONT><FONT COLOR=#FF0000>注意</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X</FONT><FONT COLOR=#FF0000>為列向量</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 10 5 -1</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> X=B*inv(A);
% </FONT><FONT COLOR=#FF0000>也可以反矩陣運(yùn)算求解<BR>
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