【线性代数】第一讲 行列式

二阶行列式的本质 - 平行四边形的面积

\[\begin{align} S_{\Box} &= l \cdot m \cdot \sin(\beta-\alpha)\\ &= l \cdot m (\sin{\beta}\cos{\alpha}-\cos{\beta}\sin{\alpha})\\ &= l\cos{\alpha}\cdot m\sin{\beta}-l\sin{\alpha}\cdot m\cos{\beta} \\ &= a_{11}\cdot a_{22} - a_{12}\cdot a_{21} \\ &= \left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right] \end{align} \]

• Cramer's Rule

  \[\begin{cases}a_{11}x_1+a_{12}x_2=b_1 \\ a_{21}x_1+a_{22}x_2=b_2 \\ \end{cases} \]

  \(x_1 = \frac{\left[ \begin{matrix} b_1 & a_{12} \\ b_2 & a_{22} \\ \end{matrix} \right]}{\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]} = \frac{b_1 a_{22}-b_2 a_{12}}{a_{11}a_{22}-a_{12}a_{21}}\)，\(x_2 = \frac{\left[ \begin{matrix} a_{11} & b_1 \\ a_{21} & b_2 \\ \end{matrix} \right]}{\left[ \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right]} = \frac{b_1 a_{22}-b_2 a_{12}}{a_{11}a_{22}-a_{12}a_{21}}\)

三阶行列式-平行六面体的体积

\[V = \left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix} \right| = (a_{11}a_{22}a_{33}+a_{21}a_{13}a_{32}+a_{31}a_{12}a_{23}) - (a_{31}a_{22}a_{13}+a_{21}a_{12}a_{33}+a_{32}a_{23}a_{11}) \]

• Cramer's Rule
  对方程组 \(\begin{cases} a_{11}x_1+a_{12}x_2+a_{13}x_3=b_1 \\ a_{21}x_1+a_{22}x_2+a_{23}x_3=b_2 \\ a_{31}x_1+a_{32}x_2+a_{33}x_3=b_3 \\ \end{cases}\)，有 \(x_j=\frac{D_j}{D}\)
  其中 $D = \left|\begin{matrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \ \end{matrix} \right| $ 为系数行列式，\(D_j\) 为将 \(D\) 中第 \(j\) 列换作 \(\left[\begin{matrix}b_1\\b_2\\b_3\\ \end{matrix}\right]\) 而得到的新行列式