MIT18.06 Linear Algebra (19)

lec19

2x2 $\det$

$$\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=\left|\begin{array}{ll}a & 0 \\c & d\end{array}\right|+\left|\begin{array}{ll}0 & b \\c & d\end{array}\right|=\left|\begin{array}{ll}a & 0 \\c & 0\end{array}\right|+\left|\begin{array}{ll}a & 0 \\0 & d\end{array}\right|+\left|\begin{array}{cc}0 & b \\c & 0\end{array}\right|+\left|\begin{array}{ll}0 & b \\0 & d\end{array}\right| = ad-bc$$

3x3 $\det$

what the survivors for

$$\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{vmatrix}$$

?

$$\begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33}\end{vmatrix}+\begin{vmatrix} a_{11} & 0 & 0\\ 0 & 0 & a_{23}\\ 0 & a_{32} & 0\end{vmatrix}+ \cdots$$

$$=a_{11} a_{22} a_{33} - a_{11} a_{23} a_{32} \cdots$$

Big formula

$$\operatorname{det} A=\sum_{n! \text { terms}} \pm a_{1\alpha} a_{2 \beta} a_{3\gamma} \cdots a_{n \omega}$$

where $(\alpha, \beta, \gamma,\ldots, \omega)=\text { perm. of }(1,2, \ldots, n)$

cofactor

$$⁍$$

子式和余子式 - 维基百科，自由的百科全书