一 引例
求解二元一次方程组
{
a
11
x
1
+
a
12
x
2
=
b
1
a
21
x
1
+
a
22
x
2
=
b
2
\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 b1