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<TITLE> 反矩阵、矩阵秩与行列式 </TITLE>
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<H1>5.3.1 反矩阵、矩阵秩与行列式</H1>
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一个正方矩阵<FONT FACE="Times New Roman">A</FONT>的反矩阵的定义是<IMG SRC="img00001-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00001.gif">，所以此二矩阵相乘不论是<IMG SRC="img00002-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00002.gif">或<IMG SRC="img00003-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00003.gif">，结果皆为单位矩阵。但是一
矩阵如果是奇异<FONT FACE="Times New Roman">(singular) </FONT>或是条件不足<FONT FACE="Times New Roman"> (ill-conditioned)</FONT>，其反矩阵并不存在。条件不足的矩阵与一组联立方程
组其中的方程式并不独立有关，而一矩阵的秩<FONT FACE="Times New Roman">(rank) </FONT>即是代表矩阵中独立方程式个数。如果一矩阵的秩数和
其矩阵的列数相等，则此矩阵为非奇异且其反矩阵存在。
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MATLAB的反矩阵函数和秩函数语法分别为<FONT COLOR=#FF0000 FACE="Times New Roman">inv(A)</FONT><TT><FONT FACE="Courier New">,
</FONT></TT><FONT COLOR=#FF0000 FACE="Times New Roman">rank(A)</FONT>，：例如：
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A=[2 1; 4
3];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> rank(A)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 2 % </FONT><FONT COLOR=#FF0000>表示</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT><FONT COLOR=#FF0000>秩数为</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">2</FONT><FONT COLOR=#FF0000>且等于矩阵的列数</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> inv(A)
% </FONT><FONT COLOR=#FF0000>反矩阵</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1.5000 -0.5000</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -2.0000 1.0000</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> B=[2 1; 3
2; 4 5]; % B</FONT><FONT COLOR=#FF0000>为奇异矩阵</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> rank(B)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 2 % </FONT><FONT COLOR=#FF0000>表示</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">B</FONT><FONT COLOR=#FF0000>秩数为</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">2</FONT><FONT COLOR=#FF0000>，但是其列数为</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">3</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> inv(B)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">??? Error using ==>
inv</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">Matrix must be square.
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相信大家都会计算矩阵行列式的值，但是如一矩阵大小超过<FONT FACE="Times New Roman">4</FONT>以上，行列式值的计算就会繁复。<FONT FACE="Times New Roman">MATLAB</FONT>提供
计算行列式的函数，其语法为<FONT COLOR=#FF0000 FACE="Times New Roman">det(A)</FONT>，例如：
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A=[1 3 0;
-1 5 2; 1 2 1];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> det(A)
% </FONT><FONT COLOR=#FF0000>矩阵之行列式值</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 10<BR>
</FONT>沪p算就会繁复。<FONT FACE="Times New Roman">MATLAB</FONT>提供
计算行列式的函数，其语法为<FONT COLOR=#FF0000 FACE="Times New Roman">det(A)</FONT>，例如：
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A=[1 3 0;
-1 5 2; 1 2 1];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> det(A)
% </FONT><FONT COLOR=#FF0000>矩阵之行列式值</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 10<BR>
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