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线性代数

萌宠菠菠乐园
2024-09-15 08:21

行列式

行列式概念的引进

先来看一个方程组:

a 11 x 1 + a 12 x 2 = b 1 , a 21 x 1 + a 22 x 2 = b 2 a_{11}x_1 + a_{12}x_2 = b_1, \a_{21}x_1 +a_{22}x_2=b_2 a11x1+a12x2=b1,a21x1+a22x2=b2

假设此方程组有解.即: a 11 a 22 − a 12 a 21 ≠ 0. a_{11}a_{22}-a_{12}a_{21} neq 0. a11a22−a12a21=0.求 x 1 和 x 2 x_1和x_2 x1和x2

一般会用高斯消元法求解.第一个方程乘上 a 22 a_{22} a22,第二个方程乘上 a 12 a_{12} a12, 两个方程相减,消掉 x 2 x_2 x2得到 x 1 x_1 x1的表达式.

x 1 = a 22 b 1 − a 12 b 2 a 11 a 2 2 − a 12 a 21 x_1 = frac{a_{22}b_1 - a_{12}b_2}{a_{11}a_22-a_{12}a_{21}} x1=a11a22−a12a21a22b1−a12b2

同理可得:

x 2 = a 11 b 2 − a 21 b 1 a 11 a 2 2 − a 12 a 21 x_2 = frac{a_{11}b_2 - a_{21}b_1}{a_{11}a_22-a_{12}a_{21}} x2=a11a22−a12a21a11b2−a21b1

能否 10秒内把上面两个公式记住?

为了记忆, 我们引进记号

约定 ∣ a 11 a 12 a 12 a 22 ∣ = a 11 a 2 2 − a 12 a 21 , 这个式子就叫做行列式约定 left|

a11a12a12a22" role="presentation">a11a12a12a22

right| = a_{11}a_22-a_{12}a_{21},这个式子就叫做行列式约定∣∣∣∣a11a12a12a22∣∣∣∣=a11a22−a12a21,这个式子就叫做行列式

行列式其实就是速记的符号,上面的行列式是二阶行列式.

这样 x 1 , x 2 x_1, x_2 x1,x2就有了新的表达式

x 1 = ∣ b 1 a 12 b 2 a 22 ∣ ∣ a 11 a 12 a 12 a 22 ∣ , x 2 = ∣ a 11 b 1 a 21 b 2 ∣ ∣ a 11 a 12 a 12 a 22 ∣ x_1 = frac {left|

b1a12b2a22" role="presentation">b1a12b2a22

right|}{left|

a11a12a12a22" role="presentation">a11a12a12a22

right|}, quad x_2= frac{left|

a11b1a21b2" role="presentation">a11b1a21b2

right|}{left|

a11a12a12a22" role="presentation">a11a12a12a22

right|} x1=∣∣∣∣a11a12a12a22∣∣∣∣∣∣∣∣b1a12b2a22∣∣∣∣,x2=∣∣∣∣a11a12a12a22∣∣∣∣∣∣∣∣a11b1a21b2∣∣∣∣

再来看一下三阶行列式:

a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 , a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 , a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3. a_{11}x_1 +a_{12}x_2 + a_{13}x_3=b_1, \a_{21}x_1+a_{22}x_2+a_{23}x_3=b2,\a_{31}x_1 + a_{32}x_2 +a_{33}x_3=b3. a11x1+a12x2+a13x3=b1,a21x1+a22x2+a23x3=b2,a31x1+a32x2+a33x3=b3.

使用高斯消元法得到 x 1 , x 2 , x 3 的分母为 a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 13 a 22 a 31 − a 11 a 23 a 32 − a 12 a 21 a 33 使用高斯消元法得到x_1, x_2, x_3的分母为a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} 使用高斯消元法得到x1,x2,x3的分母为a11a22a33+a12a23a31+a13a21a32−a13a22a31−a11a23a32−a12a21a33

我们定义以下式子为三阶行列式.

∣ a 11 a 12 a 13 a 12 a 22 a 23 a 31 a 32 a 33 ∣ = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 13 a 22 a 31 − a 11 a 23 a 32 − a 12 a 21 a 33 left|

a11a12a13a12a22a23a31a32a33" role="presentation">a11a12a13a12a22a23a31a32a33

right| = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} ∣∣∣∣∣∣a11a12a13a12a22a23a31a32a33∣∣∣∣∣∣=a11a22a33+a12a23a31+a13a21a32−a13a22a31−a11a23a32−a12a21a33

练习计算以下矩阵的行列式的值:

∣ 1 4 3 − 5 2 1 3 6 1 ∣ left|

143−521361" role="presentation">143−521361

right| ∣∣∣∣∣∣143−521361∣∣∣∣∣∣

∣ 1 0 0 − 5 2 3 3 3 5 ∣ left|

100−523335" role="presentation">100−523335

right| ∣∣∣∣∣∣100−523335∣∣∣∣∣∣

n阶行列式

D = ∣ a 11 a 12 a 13 a 12 a 22 a 23 a 31 a 32 a 33 ∣ = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 13 a 22 a 31 − a 11 a 23 a 32 − a 12 a 21 a 33 D=left|

a11a12a13a12a22a23a31a32a33" role="presentation">a11a12a13a12a22a23a31a32a33

right| = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} D=∣∣∣∣∣∣a11a12a13a12a22a23a31a32a33∣∣∣∣∣∣=a11a22a33+a12a23a31+a13a21a32−a13a22a31−a11a23a32−a12a21a33

= a 11 ( a 22 a 33 − a 23 a 32 ) + a 12 ( a 23 a 31 − a 21 a 33 ) + a 13 ( a 21 a 32 − a 22 a 31 ) =a_{11}(a_{22}a_{33} - a_{23}a_{32}) +a_{12}(a_{23}a_{31} - a_{21}a_{33}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) =a11(a22a33−a23a32)+a12(a23a31−a21a33)+a13(a21a32−a22a31)

= a 11 ∣ a 22 a 23 a 32 a 33 ∣ − a 12 ∣ a 21 a 23 a 31 a 33 ∣ + a 13 ∣ a 21 a 22 a 31 a 32 ∣ =a_{11}left|

a22a23a32a33" role="presentation">a22a23a32a33

right| - a_{12}left|

a21a23a31a33" role="presentation">a21a23a31a33

right| + a_{13}left|

a21a22a31a32" role="presentation">a21a22a31a32

right| =a11∣∣∣∣a22a23a32a33∣∣∣∣−a12∣∣∣∣a21a23a31a33∣∣∣∣+a13∣∣∣∣a21a22a31a32∣∣∣∣

= a 11 ( − 1 ) 1 + 1 ∣ a 22 a 23 a 32 a 33 ∣ + a 12 ( − 1 ) 1 + 2 ∣ a 21 a 23 a 31 a 33 ∣ + a 13 ( − 1 ) 1 + 3 ∣ a 21 a 22 a 31 a 32 ∣ =a_{11}(-1)^{1+1}left|

a22a23a32a33" role="presentation">a22a23a32a33

right| + a_{12}(-1)^{1+2}left|

a21a23a31a33" role="presentation">a21a23a31a33

right| + a_{13}(-1)^{1 + 3}left|

a21a22a31a32" role="presentation">a21a22a31a32

right| =a11(−1)1+1∣∣∣∣a22a23a32a33∣∣∣∣+a12(−1)1+2∣∣∣∣a21a23a31a33∣∣∣∣+a13(−1)1+3∣∣∣∣a21a22a31a32∣∣∣∣

记 A 11 = ( − 1 ) 1 + 1 ∣ a 22 a 23 a 32 a 33 ∣ , A 12 = ( − 1 ) 1 + 2 ∣ a 21 a 23 a 31 a 33 ∣ , A 13 = ( − 1 ) 1 + 3 ∣ a 21 a 22 a 31 a 32 ∣ A_{11} = (-1)^{1+1}left|

a22a23a32a33" role="presentation">a22a23a32a33

right|, A_{12}=(-1)^{1+2}left|

a21a23a31a33" role="presentation">a21a23a31a33

right|, A_{13}=(-1)^{1 + 3}left|

a21a22a31a32" role="presentation">a21a22a31a32

right| A11=(−1)1+1∣∣∣∣a22a23a32a33∣∣∣∣,A12=(−1)1+2∣∣∣∣a21a23a31a33∣∣∣∣,A13=(−1)1+3∣∣∣∣a21a22a31a32∣∣∣∣

D = a 11 A 11 + a 12 A 12 + a 13 A 13 D = a_{11}A_{11} +a_{12}A_{12} + a_{13}A_{13} D=a11A11+a12A12+a13A13

类似地有 D = a i 1 A i 1 + a i 2 A i 2 + a i 3 A i 3 , i = 1 , 2 , 3 D = a_{i1}A_{i1} +a_{i2}A_{i2} + a_{i3}A_{i3}, i=1,2,3 D=ai1Ai1+ai2Ai2+ai3Ai3,i=1,2,3

或 D = a 1 j A 1 j + a 2 j A 2 j + a 3 j A 3 j , j = 1 , 2 , 3 D = a_{1j}A_{1j} +a_{2j}A_{2j} + a_{3j}A_{3j}, j=1,2,3 D=a1jA1j+a2jA2j+a3jA3j,j=1,2,3

A i j 称为元素 a i j 的代数余子式 A_{ij}称为元素a_{ij}的代数余子式 Aij称为元素aij的代数余子式