一 引例
求解二元一次方程组
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\begin{cases} a_{11}x_1+a_{12}x_2=b_1\\ a_{21}x_1+a_{22}x_2=b_2\\ \end{cases}
{a11x1+a12x2=b1a21x1+a22x2=b2
解： 1 × a 21 − 2 × a 11 ⇒ x 2 = a 11 b 2 − a 21 b 1 a 11 a 22 − a 12 a 21 x 1 = a 22 b 1 − a 12 b 2 a 11 a 22 − a 12 a 21 解：\\ 1\times a_{21}-2\times a_{11}\Rightarrow\\ x_2=\frac{a_{11}b_2-a_{21}b_1}{a_{11}a_{22}-a_{12}a_{21}}\\ x_1=\frac{a_{22}b_1-a_{12}b_2}{a_{11}a_{22}-a_{12}a_{21}}\\ 解：1×a21−2×a11⇒x2=a11a22−a12a21a11b2−a21b1x1=a11a22−a12a21a22b1−a12b2
a 11 a 12 a 21 a 22 \begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix} a11a21a12a22 两行两列数表。
定义：表达式 a 11 a 22 − a 12 a 21 = Δ ∣ a 11 a 12 a 21 a 22 ∣ a_{11}a_{22}-a_{12}a_{21} \overset{\Delta}{=} \begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{vmatrix} a11a22−a12a21=Δ a11a21a12a22 为数表所确定的二阶行列式。
注：
- ∣ a 11 a 12 a 21 a 22 ∣ \begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{vmatrix} a11a21a12a22 , a i j , i = 1 , 2 , j = 1 , 2 a_{ij},i=1,2,j=1,2 aij,i=1,2,j=1,2为行列式的元素。i为元素所在的行，j位元素所在的列。
二 计算
a 11 a 22 − a 12 a 21 = ∣ a 11 a 12 a 21 a 22 ∣ = D a_{11}a_{22}-a_{12}a_{21}= \begin{vmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\\ \end{vmatrix} =D a11a22−a12a21= a11a21a12a22 =D
b 1 a 22 − b 2 a 12 = ∣ b 1 a 12 b 2 a 22 ∣ = D 1 b_1a_{22}-b_2a_{12}= \begin{vmatrix} b_1&a_{12}\\ b_2&a_{22}\\ \end{vmatrix} =D_1 b1a22−b2a12= b1b2a12a22 =D1
a 21 b 1 − a 11 b 2 = ∣ a 21 a 11 b 2 b 1 ∣ = D 2 a_{21}b_1-a_{11}b_2= \begin{vmatrix} a_{21}&a_{11}\\ b_2&b_1\\ \end{vmatrix} =D_2 a21b1−a11b2= a21b2a11b1 =D2
一种二元一次方程组的解为 x 1 = D 1 D , x 2 = D 2 D x_1=\frac{D_1}{D},x_2=\frac{D_2}{D} x1=DD1,x2=DD2
例
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\begin{cases}3x_1-2x_2=12\\2x_1+x_2=1\end{cases}
{3x1−2x2=122x1+x2=1解
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x_1=\frac{12-(-2)}{3-(-4)}=2\\ x_2=\frac{3-24}{7}=-3
x1=3−(−4)12−(−2)=2x2=73−24=−3
三 二阶行列式的几何意义
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则平行四边形的面积：
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a_{11}a_{22}-a_{12}a_{21}= \begin{vmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\\ \end{vmatrix}\\ 令\vert\vec a\vert=l,\vert\vec b\vert=m,则 平行四边形的面积：\\ S=\vec a\times\vec b=l\cdot m\cdot\sin(\beta-\alpha)\\ =l\cdot m(\sin\beta\cos\alpha-\cos\beta\sin\alpha)\\ =l\cos\alpha\cdot m\sin\beta-l\sin\alpha\cdot m\cos\beta\\ =a_{11}a_{22}-a_{12}a_{21}
a11a22−a12a21=
a11a21a12a22
令∣a∣=l,∣b∣=m,则平行四边形的面积：S=a×b=l⋅m⋅sin(β−α)=l⋅m(sinβcosα−cosβsinα)=lcosα⋅msinβ−lsinα⋅mcosβ=a11a22−a12a21
上述二阶行列式几何意义：以行向量(第一行为向量和以第二行为向量)为邻边所构成平行四边形的面积。
三阶行列式放在后面N阶行列式讲解。
结语
❓QQ:806797785
⭐️文档笔记地址：https://gitee.com/gaogzhen/math
参考：
[1]同济六版《线性代数》全程教学视频[CP/OL].2020-02-07.p2.
