线性代数第一章知识点总结篇
行列式定义
逆序数
1.n!个n排列中，惟有123...(n–1)n是按顺序排列的，称该排列为标准排列
2.任意排列通过一次变换必将改变奇偶性
3.一段排列的逆序总数称为逆序数，例如:15243的逆序数为4，记τ(15243)=4
(为了方便，如下用\(\tau\)标识)
4.逆序数为偶数的排列称为偶排列，一样，逆序数为奇数的排列称为奇排列
spa
二阶行列式计算
\[\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \]
\[=\sum_{j_1j_2}(-1)^{\tau(j_1j_2)}a_{1j1}a_{2j2}= a*d-b*c \]
三阶行列式计算
\[\left| \begin{matrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{matrix} \right| \]
\[=\sum_{j_1j_2j_3}-1^{\tau(j_1j_2j_3)}a_{1j_1}a_{2j_2}a_{3j_3} = a*e*i+b*f*g+c*d*h-c*e*g-f*h*a-i*b*d \]
对角线法则class
上下三角形
\[\left| \begin{matrix} a & * & *\\ 0 & e & *\\ 0 & 0 & i\\ \end{matrix} \right|=上三角 \]
\[\left| \begin{matrix} a & 0 & 0\\ x & e & 0\\ x & x & i\\ \end{matrix} \right| = 下三角 \]
注意点(杂项)
1.行列式中，行列地位平等
(证实请看参考下篇的证实习题篇)
2.一阶行列式 |a|=a
3.代数和中每一项的正负号的决定方法：当行指标取成标准排列时，由列指标组成的排列的奇偶性肯定，偶者为正，奇者为负
见上面的\(\tau(j_1j_2j_3)\)方法
行列式性质
1.D=DT(行列式行列互换，值不变)
证实：总结
\[D= \left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ b_{21} & b_{22} & ... & b_{2n}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
\[= \sum_{j_1j_2j_3}(-1)^{\tau(j_1j_2j_3)}b_{1j_1}b_{2j_2}b_{3j_3}…b_{nj_n} \]
\[= \sum_{j_1j_2j_3}(-1)^{\tau(j_1j_2j_3)}b_{j_11}b_{j_22}b_{j_33}…b_{nj_n} \]
\[= \left| \begin{matrix} b_{11} & b_{21} & ... & b_{n1}\\ b_{12} & b_{22} & ... & b_{n2}\\ ... & ... & & ...\\ b_{1n} & b_{2n} & ... & b_{nn}\\ \end{matrix} \right| \]
2.行列式中两行(列)交换一次，行列式的值增长一个负号。
i行和j行交换:di
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ —&—&—&— \\ |b_{i1} & b_{i2} & ... & b_{in}|\\ |... & ... & & ...|\\ |b_{j1} & b_{j2} & ... & b_{jn}|\\ —&—&—&— \\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = \]
\[- \left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ —&—&—&— \\ |b_{j1} & b_{j2} & ... & b_{jn}|\\ |... & ... & & ...|\\ |b_{i1} & b_{i2} & ... & b_{in}|\\ —&—&—&— \\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
推1.行列式两行(列)对应元素全相等，则行列式为零。
display
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ —&—&—&— \\ |b_{j1} & b_{j2} & ... & b_{jn}|\\ |... & ... & & ...|\\ | b_{j1} & b_{j2} & ... & b_{jn}|\\ —&—&—&— \\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = 0 \]
3.K倍行列式=行列式某一行(列)的K倍
math
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ kb_{21} & kb_{22} & ... & kb_{2n}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = \]
\[k \left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ b_{21} & b_{22} & ... & b_{2n}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
推2.行列式中某一行元素全为0则行列式的值为0。
play
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ 0 & 0 & ... & 0\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = 0 \]
推3.行列式中某一行(列)元素与另外一行(列)元素成比例，则行列式的值为0。
4.行列式中第i行(列)的k倍加(减)第j行(列)的m倍，行列式的值不变。
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ —&—&—&— \\ |b_{i1} & b_{i2} & ... & b_{in}|\\ |... & ... & & ...|\\ | b_{j1} & b_{j2} & ... & b_{jn}|\\ —&—&—&— \\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = \]
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ mb_{i1}+nb_{j1} & mb_{i2}+nb_{j2} & ... & mb_{in}+nb_{jn}\\ ... & ... & & ...\\ b_{j1} & b_{j2} & ... & b_{jn}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
5.行列式分行相加性
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ b_{i1}+c_{i1} & b_{i2}+c_{i2} & ... & b_{in}+c_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = \]
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ b_{i1} & b_{i2} & ... & b_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| + \]
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ c_{i1} & c_{i2} & ... & c_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
推广到n：
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ b_{i1}+c_{i1}+...+k_{i1} & b_{i2}+c_{i2}+...+k_{i2} & ... & b_{in}+c_{in}+...+k_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| = \]
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ b_{i1} & b_{i2} & ... & b_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| + \]
\[\left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ c_{i1} & c_{i2} & ... & c_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right|+... \]
\[+ \left| \begin{matrix} b_{11} & b_{12} & ... & b_{1n}\\ ... & ... & & ...\\ k_{i1} & k_{i2} & ... & k_{in}\\ ... & ... & & ...\\ b_{n1} & b_{n2} & ... & b_{nn}\\ \end{matrix} \right| \]
行列式按行或列展开(降阶)
1.余子式：去掉元素\(a_{ij}\)所在的第i行第j列的元素，留下的元素按原来位置构成的n-1阶行列式为元素\(a_{ij}\)的余子式，记：\(M_{ij}\)
2.代数余子式：\(A_{ij} = (-1)^{i+j}M_{ij}\)
\[D= \left| \begin{matrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23}\\ b_{31} & b_{32}& b_{33}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[M_{32}= \left| \begin{matrix} b_{11} & b_{13}\\ b_{21} & b_{23}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[A_{32} = (-1)^{3+2}M_{32} \]
<Empty \space Math \space Block>
\[=(-1)^5M_{32}=-M_{32} \]
3.n阶行列式D = |\(a_{ij}\)|等于它的任意一行（列）的各元素与其对应的代数余子式（\(A_{ij}\)）乘积之和
规则:利用行列式的性质，使某一行（列）尽量地有较多的0，再按行列展开！
例：
\[D= \left| \begin{matrix} -1 & 1 & -1 & 2\\ 1 & 0 & 1 & -1\\ 2 & 4 & 3 & 1\\ -1 & 1 & 2 & -2\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[ = \left| \begin{matrix} 1 & 1 & 1 & 2\\ 0 & 0 & 0 & -1\\ 3 & 4 & 4 & 1\\ -3 & 1 & 0 & -2\\ \end{matrix} \right| = (-1)^{2+4}(-1)* \left| \begin{matrix} 1 & 1 & 1 \\ 3 & 4 & 4 \\ -3 & 1 & 0\\ \end{matrix} \right| = 1 \]
克拉默法则
线性方程组的系数 矩阵 行列式\(D≠0\),则行列式有惟一解，且解为：
\[x_1 = \frac {D_1}{D},x_2 = \frac {D_2}{D},x_3 = \frac {D_3}{D} \]
八个基本型
上三角:
\[\left| \begin{matrix} b_{11} & 0 & 0 & 0\\ * & b_{22} & 0 & 0\\ * & *& b_{33} & 0\\ ... & ... & & ...\\ *& * &*& b_{nn}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[ = b_{11}*b_{22}*b_{33}*...*b_{nn} \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} b_{11} & * & * & *\\ 0 & b_{22} & * & *\\ 0 & 0& b_{33} & *\\ ... & ... & & ...\\ 0& 0 &0& b_{nn}\\ \end{matrix} \right| \]
下三角:
\[\left| \begin{matrix} 0& 0 & 0 & b_{1n}\\ 0 & 0 & b_{2(n-1)} & *\\ 0 & b_{3(n-2)} & *& *\\\\ ... & ... & .... & ...\\ b_{n1}& * &*&*\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} *& * & * & b_{1n}\\ * & * & b_{2(n-1)} & 0\\ * & b_{3(n-2)} & 0& 0\\\\ ... & ... & .... & ...\\ b_{n1}& 0 &0&0\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[ = (-1)^{\frac {n(n-1)}{2}}b_{1(n-1)}*b_{2(n-2)}*b_{3(n-3)}*...*b_{1n} \]
组合三角:
\[\left| \begin{matrix} b_{11} & b_{12} & 0 & 0\\ b_{21} & b_{22} & 0 & 0\\ * & *&c_{11} & c_{12}\\ *& * &c_{21}& c_{22}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[ = \left| \begin{matrix} b_{11} & b_{12} & * & *\\ b_{21} & b_{22} & * & *\\ 0 & 0& c_{11} & c_{12}\\ 0& 0 &c_{21}& c_{22}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{matrix} \right|* \left| \begin{matrix} c_{11} & c_{12}\\ c_{21}& c_{22}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[\left| \begin{matrix} 0 & 0&b_{11} & b_{12} \\ 0 & 0&b_{21} & b_{22}\\ c_{11} & c_{12}&* & *\\ c_{21}& c_{22}&*& * \\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} * & * &b_{11} & b_{12}\\ * & *& b_{21} & b_{22}\\ c_{11} & c_{12}&0 & 0 \\ c_{21}& c_{22}&0& 0 \\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= (-1)^{2*2} \left| \begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{matrix} \right|* \]
<Empty \space Math \space Block>
\[\left| \begin{matrix} c_{11} & c_{12}\\ c_{21}& c_{22}\\ \end{matrix} \right| \]
范德蒙行列式
\[D=(a_1,a_2,...,a_n) \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} 1& 1&... & 1\\ a_1& a_2&... & a_n\\ a_1^2& a_2^2&... & a_n^2\\ ... & ... & ... & ...\\ a_1^{(n-1)}& a_2^{(n-1)}&... & a_n^{(n-1)}\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \prod_{1≤i<j≤n}(a_j-a_i) \]
重要行列式解法
type1:
\[f(x)= \left| \begin{matrix} x& a_1&a_2&...&a_{n-1} & 1\\ a_1&x& a_2&...&a_{n-1} & 1\\ a_1& a_2&x&...&a_{n-1} & 1\\ ... & ... & ... & ...& ... & ...\\ a_1& a_2&a_3&... & x&1\\ a_1& a_2&a_3&... & a_n&1\\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} x-a_1& a_1-x&0&...&0 & 0\\ 0&x-a_2& a_2-x&...&0 & 0\\ 0& 0&x-a_3&...&0& 0\\ ... & ... & ... & ...& ... & ...\\ 0& 0&0&... & x-a_n&0\\ a_1& a_2&a_3&... & a_n&1\\ \end{matrix} \right| (列展开) \]
<Empty \space Math \space Block>
\[=\left| \begin{matrix} x-a_1& a_1-x&0&...&0 \\ 0&x-a_2& a_2-x&...&0 \\ 0& 0&x-a_3&...&0\\ ... & ... & ... & ...& ... \\ 0& 0&0&... & x-a_n\\ \end{matrix} \right| = (x-a_1)(x-a_2)...(x-a_n) \]
type2:
\[f(x)= \left| \begin{matrix} a& b& b&...& b\\ b&a& b&...&b \\ b&b& a&...&b \\ ... & ... & ... & ...& ...\\ b&b& b&...&a \\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[= \left| \begin{matrix} a+(n-1)b& b& b&...& b\\ a+(n-1)b&a& b&...&b \\ a+(n-1)b&b& a&...&b \\ ... & ... & ... & ...& ...\\ a+(n-1)b&b& b&...&a \\ \end{matrix} \right| (提取a+(n-1)b) \]
<Empty \space Math \space Block>
\[=[a+(n-1)b] \left| \begin{matrix} 1 & b& b&...& b\\ 1&a& b&...&b \\ 1&b& a&...&b \\ ... & ... & ... & ...& ...\\ 1&b& b&...&a \\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[=[a+(n-1)b] \left| \begin{matrix} 1 & b& b&...& b\\ 0&a-b& 0&...&0 \\ 0&0& a-b&...&0 \\ ... & ... & ... & ...& ...\\ 0&0& 0&...&a-b \\ \end{matrix} \right| \]
<Empty \space Math \space Block>
\[ = [a+(n-1)b](a-b)_{n-1} \]