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<TITLE> 反矩陣、矩陣秩與行列式 </TITLE>
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<H1>5.3.1 反矩陣、矩陣秩與行列式</H1>
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一個正方矩陣<FONT FACE="Times New Roman">A</FONT>的反矩陣的定義是<IMG SRC="img00001-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00001.gif">，所以此二矩陣相乘不論是<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">，結果皆為單位矩陣。但是一
矩陣如果是奇異<FONT FACE="Times New Roman">(singular) </FONT>或是條件不足<FONT FACE="Times New Roman"> (ill-conditioned)</FONT>，其反矩陣并不存在。條件不足的矩陣與一組聯立方程
組其中的方程式并不獨立有關，而一矩陣的秩<FONT FACE="Times New Roman">(rank) </FONT>即是代表矩陣中獨立方程式個數。如果一矩陣的秩數和
其矩陣的列數相等，則此矩陣為非奇異且其反矩陣存在。
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MATLAB的反矩陣函數和秩函數語法分別為<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>秩數為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">2</FONT><FONT COLOR=#FF0000>且等于矩陣的列數</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> inv(A)
% </FONT><FONT COLOR=#FF0000>反矩陣</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>為奇異矩陣</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>秩數為</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">2</FONT><FONT COLOR=#FF0000>，但是其列數為</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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相信大家都會計算矩陣行列式的值，但是如一矩陣大小超過<FONT FACE="Times New Roman">4</FONT>以上，行列式值的計算就會繁復。<FONT FACE="Times New Roman">MATLAB</FONT>提供
計算行列式的函數，其語法為<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>矩陣之行列式值</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>滬p算就會繁復。<FONT FACE="Times New Roman">MATLAB</FONT>提供
計算行列式的函數，其語法為<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>矩陣之行列式值</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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