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<TITLE> 利用矩阵解法 </TITLE>
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<H1>6.1 利用矩阵解法</H1>
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假设一组联立线性方程式为
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我们习惯将上组方程式以矩阵方式表示如下
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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>为等式左边各方程式的系数项，<FONT FACE="Times New Roman">X
</FONT>为欲求解的未知项，<FONT FACE="Times New Roman">B </FONT>代表等式右边之已知项
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要解上述的联立方程式，我们可以利用在第六章介绍的矩阵左除<FONT FACE="Times New Roman">
</FONT><FONT COLOR=#FF0000>\</FONT><FONT FACE="Times New Roman">
</FONT>做运算，即是<FONT FACE="Times New Roman"> </FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=A\B</FONT>。
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如果将原方程式改写成<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>分别为
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注意上式的<FONT FACE="Times New Roman"> X, B </FONT>已改写成列向量，<FONT FACE="Times New Roman">A</FONT>其实是前一个方程式中<FONT FACE="Times New Roman">
A </FONT>的转置矩阵。上式的<FONT FACE="Times New Roman"> X </FONT>可以矩阵右除<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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若以反矩阵运算求解<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>，或是改写成<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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我们直接以下面的例子来说明这三个运算的用法：
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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>将等式的左边系数键入</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> B=[10 5 -1]';
% </FONT><FONT COLOR=#FF0000>将等式右边之已知项键入，</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">B</FONT><FONT COLOR=#FF0000>要做转置</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> X=A\B
% </FONT><FONT COLOR=#FF0000>先以左除运算求解</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>为行向量</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>验算解是否正确</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>将</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT><FONT COLOR=#FF0000>先做转置</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>以右除运算求解的结果亦同</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>为列向量</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>也可以反矩阵运算求解<BR>
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