考研数学之线性代数知识点整理——1.行列式
本系列博客汇总在这里：考研数学知识点汇总系列博客
文章目录
- 一、行列式
- 1 行列式的定义
- 2 行列式的展开
- 2.1 余子式和代数余子式
- 2.2 行列式按行(列)展开
- 3 行列式的性质
- 4 拉普拉斯定理与范德蒙德行列式
- 4.1 拉普拉斯定理
- 4.2 范德蒙德行列式
- 5 克拉默法则
一、行列式
1 行列式的定义
n阶行列式
∣ a 11 a 12 ⋅ ⋅ ⋅ a 1 n a 21 a 22 ⋅ ⋅ ⋅ a 2 n ⋮ ⋮ ⋮ a n 1 a n 2 ⋅ ⋅ ⋅ a n n ∣ = ∑ j 1 j 2 ⋅ ⋅ ⋅ j n ( − 1 ) τ ( j 1 j 2 ⋅ ⋅ ⋅ j n ) a 1 j 1 a 2 j 2 ⋅ ⋅ ⋅ a n j n \begin{vmatrix} a_{11} & a_{12} & ··· & a_{1n} \\ a_{21} & a_{22} & ··· & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & ··· & a_{nn} \end{vmatrix} = \sum\limits_{j_1j_2···j_n}(-1)^{τ(j_1j_2···j_n)}a_{1j_1}a_{2j_2}···a_{nj_n} ∣∣∣∣∣∣∣∣∣a11a21⋮an1a12a22⋮an2⋅⋅⋅⋅⋅⋅⋅⋅⋅a1na2n⋮ann∣∣∣∣∣∣∣∣∣=j1j2⋅⋅⋅jn∑(−1)τ(j1j2⋅⋅⋅jn)a1j1a2j2⋅⋅⋅anjn
其中 j 1 j 2 ⋅ ⋅ ⋅ j n j_1j_2···j_n j1j2⋅⋅⋅jn是1,2,···,n的一个排列， ∑ j 1 j 2 ⋅ ⋅ ⋅ j n \sum\limits_{j_1j_2···j_n} j1j2⋅⋅⋅jn∑表示对所有的n级排列求和。
2 行列式的展开
2.1 余子式和代数余子式
在n阶行列式 D = ∣ a i j ∣ D=|a_{ij}| D=∣aij∣中划去元素 a i j a_{ij} aij所在的第i行与第j行，剩下的元素按原来的排法构成一个n-1阶行列式，称为元素 a i j a_{ij} aij的余子式，记为 M i j M_{ij} Mij，称 A i j = ( − 1 ) i + j M i j A_{ij}=(-1)^{i+j}M_{ij} Aij=(−1)i+jMij为元素 a i j a_{ij} aij的代数余子式。
2.2 行列式按行(列)展开
D = a i 1 A i 1 + a i 2 A i 2 + ⋅ ⋅ ⋅ + a i n A i n ( 1 ⩽ i ⩽ n ) D=a_{i1}A_{i1}+a_{i2}A_{i2}+···+a_{in}A_{in}(1 \leqslant i \leqslant n) D=ai1Ai1+ai2Ai2+⋅⋅⋅+ainAin(1⩽i⩽n)
D = a 1 j A 1 j + a 2 j A 2 j + ⋅ ⋅ ⋅ + a n j A n j ( 1 ⩽ j ⩽ n ) D=a_{1j}A_{1j}+a_{2j}A_{2j}+···+a_{nj}A_{nj}(1 \leqslant j \leqslant n) D=a1jA1j+a2jA2j+⋅⋅⋅+anjAnj(1⩽j⩽n)
∑ i = 1 n a i j A i k = a 1 j A 1 k + a 2 j A 2 k + ⋅ ⋅ ⋅ + a n j A n k = { D , j = k 0 , j ≠ k \sum\limits_{i=1}^{n}a_{ij}A_{ik}=a_{1j}A_{1k}+a_{2j}A_{2k}+···+a_{nj}A_{nk}=\begin{cases} D,j=k \\ 0,j≠k \end{cases} i=1∑naijAik=a1jA1k+a2jA2k+⋅⋅⋅+anjAnk={D,j=k0,j̸=k
3 行列式的性质
① 经转置行列式的值不变，即 ∣ D T ∣ = ∣ D ∣ |D^T|=|D| ∣DT∣=∣D∣。
② 互换行列式的两行(列)，行列式变号。
∣ … … … … a i 1 a i 2 … a i n … … … … a j 1 a j 2 … a j n … … … … ∣ = − ∣ … … … … a j 1 a j 2 … a j n … … … … a i 1 a i 2 … a i n … … … … ∣ \begin{vmatrix} \dots & \dots & \dots & \dots \\ a_{i1} & a_{i2} & \dots & a_{in} \\ \dots & \dots & \dots & \dots \\ a_{j1} & a_{j2} & \dots & a_{jn} \\ \dots & \dots & \dots & \dots \\ \end{vmatrix} =-\begin{vmatrix} \dots & \dots & \dots & \dots \\ a_{j1} & a_{j2} & \dots & a_{jn} \\ \dots & \dots & \dots & \dots \\ a_{i1} & a_{i2} & \dots & a_{in} \\ \dots & \dots & \dots & \dots \\ \end{vmatrix} ∣∣∣∣∣∣∣∣∣∣…ai1…aj1……ai2…aj2…………………ain…ajn…∣∣∣∣∣∣∣∣∣∣=−∣∣∣∣∣∣∣∣∣∣…aj1…ai1……aj2…ai2…………………ajn…ain…∣∣∣∣∣∣∣∣∣∣
③ 行列式某一行(列)中所有元素都乘以数k，等于k乘以此行列式。
∣ k a i 1 k a i 2 … k a i n ∣ = k ∣ a i 1 a i 2 … a i n ∣ \begin{vmatrix} ka_{i1} & ka_{i2} & \dots & ka_{in} \\\end{vmatrix}=k \begin{vmatrix} a_{i1} & a_{i2} & \dots & a_{in} \\\end{vmatrix} ∣∣kai1kai2…kain∣∣=k∣∣ai1ai2…ain∣∣
④ 若行列式某两行(列)元素成比例，则行列式等于零。
∣ … … … … a i 1 a i 2 … a i n … … … … k a i 1 k a i 2 … k a i n … … … … ∣ = 0 \begin{vmatrix} \dots & \dots & \dots & \dots \\ a_{i1} & a_{i2} & \dots & a_{in} \\ \dots & \dots & \dots & \dots \\ ka_{i1} & ka_{i2} & \dots & ka_{in} \\ \dots & \dots & \dots & \dots \\ \end{vmatrix}=0 ∣∣∣∣∣∣∣∣∣∣…ai1…kai1……ai2…kai2…………………ain…kain…∣∣∣∣∣∣∣∣∣∣=0
⑤ 若行列式某一行(列)的元素都是两数之和，则行列式等于两个行列式之和。
∣ a 11 ⋅ ⋅ ⋅ a 1 j ⋅ ⋅ ⋅ a 1 n ⋮ ⋮ ⋮ a i 1 + a i 1 ′ ⋅ ⋅ ⋅ a i j + a i j ′ ⋅ ⋅ ⋅ a i n + a i n ′ ⋮ ⋮ ⋮ a n 1 ⋅ ⋅ ⋅ a n j ⋅ ⋅ ⋅ a n n ∣ = ∣ a 11 ⋅ ⋅ ⋅ a 1 j ⋅ ⋅ ⋅ a 1 n ⋮ ⋮ ⋮ a i 1 ⋅ ⋅ ⋅ a i j ⋅ ⋅ ⋅ a i n ⋮ ⋮ ⋮ a n 1 ⋅ ⋅ ⋅ a n j ⋅ ⋅ ⋅ a n n ∣ + ∣ a 11 ⋅ ⋅ ⋅ a 1 j ⋅ ⋅ ⋅ a 1 n ⋮ ⋮ ⋮ a i 1 ′ ⋅ ⋅ ⋅ a i j ′ ⋅ ⋅ ⋅ a i n ′ ⋮ ⋮ ⋮ a n 1 ⋅ ⋅ ⋅ a n j ⋅ ⋅ ⋅ a n n ∣ \begin{vmatrix} a_{11} & ··· & a_{1j} & ··· & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1}+a_{i1}' & ··· & a_{ij}+a_{ij}' & ··· & a_{in}+a_{in}' \\ \vdots & & \vdots & & \vdots \\ a_{n1} & ··· & a_{nj} & ··· & a_{nn} \end{vmatrix} =\begin{vmatrix} a_{11} & ··· & a_{1j} & ··· & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1} & ··· & a_{ij} & ··· & a_{in} \\ \vdots & & \vdots & & \vdots \\ a_{n1} & ··· & a_{nj} & ··· & a_{nn} \end{vmatrix} +\begin{vmatrix} a_{11} & ··· & a_{1j} & ··· & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1}' & ··· & a_{ij}' & ··· & a_{in}' \\ \vdots & & \vdots & & \vdots \\ a_{n1} & ··· & a_{nj} & ··· & a_{nn} \end{vmatrix} ∣∣∣∣∣∣∣∣∣∣∣∣a11⋮ai1+ai1′⋮an1⋅⋅⋅⋅⋅⋅⋅⋅⋅a1j⋮aij+aij′⋮anj⋅⋅⋅⋅⋅⋅⋅⋅⋅a1n⋮ain+ain′⋮ann∣∣∣∣∣∣∣∣∣∣∣∣=∣∣∣∣∣∣∣∣∣∣∣∣a11⋮ai1⋮an1⋅⋅⋅⋅⋅⋅⋅⋅⋅a1j⋮aij⋮anj⋅⋅⋅⋅⋅⋅⋅⋅⋅a1n⋮ain⋮ann∣∣∣∣∣∣∣∣∣∣∣∣+∣∣∣∣∣∣∣∣∣∣∣∣a11⋮ai1′⋮an1⋅⋅⋅⋅⋅⋅⋅⋅⋅a1j⋮aij′⋮anj⋅⋅⋅⋅⋅⋅⋅⋅⋅a1n⋮ain′⋮ann∣∣∣∣∣∣∣∣∣∣∣∣
⑥ 把行列式的某一行(列)的各元素乘以同一数加到另一行(列)对应元素上去，行列式的值不变。
∣ a 11 ⋅ ⋅ ⋅ a 1 j ⋅ ⋅ ⋅ a 1 n ⋮ ⋮ ⋮ a i 1 ⋅ ⋅ ⋅ a i j ⋅ ⋅ ⋅ a i n ⋮ ⋮ ⋮ a n 1 ⋅ ⋅ ⋅ a n j ⋅ ⋅ ⋅ a n n ∣ = ∣ a 11 ⋅ ⋅ ⋅ a 1 j ⋅ ⋅ ⋅ a 1 n ⋮ ⋮ ⋮ a i 1 + k a 11 ⋅ ⋅ ⋅ a i j + k a 1 j ⋅ ⋅ ⋅ a i n + k a 1 n ⋮ ⋮ ⋮ a n 1 ⋅ ⋅ ⋅ a n j ⋅ ⋅ ⋅ a n n ∣ \begin{vmatrix} a_{11} & ··· & a_{1j} & ··· & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1} & ··· & a_{ij} & ··· & a_{in} \\ \vdots & & \vdots & & \vdots \\ a_{n1} & ··· & a_{nj} & ··· & a_{nn} \end{vmatrix} =\begin{vmatrix} a_{11} & ··· & a_{1j} & ··· & a_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{i1}+ka_{11} & ··· & a_{ij}+ka_{1j} & ··· & a_{in}+ka_{1n} \\ \vdots & & \vdots & & \vdots \\ a_{n1} & ··· & a_{nj} & ··· & a_{nn} \end{vmatrix} ∣∣∣∣∣∣∣∣∣∣∣∣a11⋮ai1⋮an1⋅⋅⋅⋅⋅⋅⋅⋅⋅a1j⋮aij⋮anj⋅⋅⋅⋅⋅⋅⋅⋅⋅a1n⋮ain⋮ann∣∣∣∣∣∣∣∣∣∣∣∣=∣∣∣∣∣∣∣∣∣∣∣∣a11⋮ai1+ka11⋮an1⋅⋅⋅⋅⋅⋅⋅⋅⋅a1j⋮aij+ka1j⋮anj⋅⋅⋅⋅⋅⋅⋅⋅⋅a1n⋮ain+ka1n⋮ann∣∣∣∣∣∣∣∣∣∣∣∣
4 拉普拉斯定理与范德蒙德行列式
4.1 拉普拉斯定理
在n阶行列式D中任意取定k行(1≤k≤n)，由这k行元素组成的所有k阶子式与它们的代数余子式乘积之和等于行列式D，即
D = M 1 A 1 + M 2 A 2 + ⋅ ⋅ ⋅ + M s A s ( s = C n k ) D=M_1A_1+M_2A_2+···+M_sA_s(s=C_n^k) D=M1A1+M2A2+⋅⋅⋅+MsAs(s=Cnk)
其中 A i A_i Ai是子式 M i ( i = 1 , 2 , ⋅ ⋅ ⋅ , s ) M_i(i=1,2,···,s) Mi(i=1,2,⋅⋅⋅,s)对应的代数余子式。
4.2 范德蒙德行列式
D n = ∣ 1 1 1 ⋅ ⋅ ⋅ 1 a 1 a 2 a 3 ⋅ ⋅ ⋅ a n a 1 2 a 2 2 a 3 2 ⋅ ⋅ ⋅ a n 2 ⋮ ⋮ ⋮ ⋮ a 1 n − 2 a 2 n − 2 a 3 n − 2 ⋅ ⋅ ⋅ a n n − 2 a 1 n − 1 a 2 n − 1 a 3 n − 1 ⋅ ⋅ ⋅ a n n − 1 ∣ D_n= \begin{vmatrix} 1 & 1 & 1 & ··· & 1 \\ a_1 & a_2 & a_3 & ··· & a_n \\ {a_1}^2 & {a_2}^2 & {a_3}^2 & ··· & {a_n}^2 \\ \vdots & \vdots & \vdots & & \vdots \\ {a_1}^{n-2} & {a_2}^{n-2} & {a_3}^{n-2} & ··· & {a_n}^{n-2} \\ {a_1}^{n-1} & {a_2}^{n-1} & {a_3}^{n-1} & ··· & {a_n}^{n-1} \\ \end{vmatrix} Dn=∣∣∣∣∣∣∣∣∣∣∣∣∣1a1a12⋮a1n−2a1n−11a2a22⋮a2n−2a2n−11a3a32⋮a3n−2a3n−1⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅1anan2⋮ann−2ann−1∣∣∣∣∣∣∣∣∣∣∣∣∣
称为n阶范德蒙德行列式。n阶范德蒙德行列式等于 a 1 , a 2 , ⋅ ⋅ ⋅ , a n a_1,a_2,···,a_n a1,a2,⋅⋅⋅,an这n个数的所有可能的差 a i − a j ( 1 ⩽ j < i ⩽ n ) a_i-a_j(1\leqslant j < i \leqslant n) ai−aj(1⩽j<i⩽n)的乘积，即
D n = ∏ 1 ⩽ j < i ⩽ n ( a i − a j ) D_n=\prod\limits_{1\leqslant j<i\leqslant n} (a_i-a_j) Dn=1⩽j<i⩽n∏(ai−aj)
5 克拉默法则
若非齐次方程组
{ a 11 x 1 + a 12 x 2 + ⋅ ⋅ ⋅ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋅ ⋅ ⋅ + a 2 n x n = b 2 ⋮ ⋮ ⋮ ⋮ a n 1 x 1 + a n 2 x 2 + ⋅ ⋅ ⋅ + a n n x n = b n \begin{cases} a_{11}x_1+a_{12}x_2+···+a_{1n}x_n=b_1 \\ a_{21}x_1+a_{22}x_2+···+a_{2n}x_n=b_2 \\ \quad\vdots \qquad\quad\vdots \qquad\qquad\quad\vdots \qquad\quad\vdots\\ a_{n1}x_1+a_{n2}x_2+···+a_{nn}x_n=b_n \\ \end{cases} ⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧a11x1+a12x2+⋅⋅⋅+a1nxn=b1a21x1+a22x2+⋅⋅⋅+a2nxn=b2⋮⋮⋮⋮an1x1+an2x2+⋅⋅⋅+annxn=bn
系数行列式D=|A|≠0，则方程组有唯一解且
x 1 = D 1 D , x 2 = D 2 D , ⋅ ⋅ ⋅ , x n = D n D x_1=\frac{D_1}{D},x_2=\frac{D_2}{D},···,x_n=\frac{D_n}{D} x1=DD1,x2=DD2,⋅⋅⋅,xn=DDn
其中， D i = ∣ a 11 … a 1 i − 1 b 1 a 1 i + 1 … a 1 n a 21 … a 2 i − 1 b 2 a 2 i + 1 … a 2 n ⋮ ⋮ ⋮ ⋮ ⋮ a n 1 … a n i − 1 b n a n i + 1 … a n n ∣ D_i=\begin{vmatrix} a_{11} & \dots & a_{1i-1} & b_1 & a_{1i+1} & \dots & a_{1n} \\ a_{21} & \dots & a_{2i-1} & b_2 & a_{2i+1} & \dots & a_{2n} \\ \vdots & & \vdots & \vdots & \vdots & & \vdots \\ a_{n1} & \dots & a_{ni-1} & b_n & a_{ni+1} & \dots & a_{nn} \\ \end{vmatrix} Di=∣∣∣∣∣∣∣∣∣a11a21⋮an1………a1i−1a2i−1⋮ani−1b1b2⋮bna1i+1a2i+1⋮ani+1………a1na2n⋮ann∣∣∣∣∣∣∣∣∣
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推论 1
若齐次方程组
{ a 11 x 1 + a 12 x 2 + ⋅ ⋅ ⋅ + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + ⋅ ⋅ ⋅ + a 2 n x n = 0 ⋮ ⋮ ⋮ ⋮ a n 1 x 1 + a n 2 x 2 + ⋅ ⋅ ⋅ + a n n x n = 0 \begin{cases} a_{11}x_1+a_{12}x_2+···+a_{1n}x_n=0 \\ a_{21}x_1+a_{22}x_2+···+a_{2n}x_n=0 \\ \quad\vdots \qquad\quad\vdots \qquad\qquad\quad\vdots \qquad\quad\vdots\\ a_{n1}x_1+a_{n2}x_2+···+a_{nn}x_n=0 \\ \end{cases} ⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧a11x1+a12x2+⋅⋅⋅+a1nxn=0a21x1+a22x2+⋅⋅⋅+a2nxn=0⋮⋮⋮⋮an1x1+an2x2+⋅⋅⋅+annxn=0
的系数行列式D=|A|≠0，则方程组只有一组零解。
x 1 = 0 , x 2 = 0 , ⋅ ⋅ ⋅ , x n = 0 x_1=0,x_2=0,···,x_n=0 x1=0,x2=0,⋅⋅⋅,xn=0
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推论 2
若齐次方程组有非零解，则它的系数行列式必为0。