【数学】线性代数知识点阶段性汇总
题型一:与行列式有关的计算
行的逆序数或者列的逆序数皆可
D n = ∣ a 1 a 12 ⋯ a 1 n ⋯ ⋯ a n 1 a n 2 ⋯ a n n ∣ = Σ ( − 1 ) t ( p 1 p 2 . . . p n ) a 1 p 1 a 2 p 2 . . . a n p n = Σ ( − 1 ) t ( p 1 p 2 . . . p n ) a p 1 1 a p 2 2 . . . a p n n D_{n}=\left|\begin{array}{ccc} a_{1} a_{12} & \cdots & a_{1 n} \\ \cdots & \cdots \\ a_{n 1} a_{n 2} & \cdots & a_{n n} \end{array}\right|=\Sigma(-1) ^{t(p_1p_2...p_n)}a_{1p_1}a_{2p_2}...a_{np_n}=\Sigma(-1) ^{t(p_1p_2...p_n)}a_{p_{1} 1} a_{p_22}...a_{p_nn} Dn=∣∣∣∣∣∣a1a12⋯an1an2⋯⋯⋯a1nann∣∣∣∣∣∣=Σ(−1)t(p1p2...pn)a1p1a2p2...anpn=Σ(−1)t(p1p2...pn)ap11ap22...apnn
题型二:与行列式有关的计算或者证明
知识点
行列式的性质
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D = D T D=D^T D=DT
D = ∣ a 11 ⋯ a 1 n ⋯ a n 1 ⋯ a n n ∣ D T = ∣ a 11 ⋯ a n 1 ⋮ a 1 n ⋯ a n n ∣ D=\left|\begin{array}{ccc} a_{11} & \cdots & \operatorname{a_{1n}} \\ \cdots & & \\ a_{n 1} & \cdots & a_{n n} \end{array}\right| \quad D^T=\left|\begin{array}{ccc} a_{11} & \cdots & a_{n 1} \\ \vdots & & \\ a_{1 n} & \cdots & a_{n n} \end{array}\right| D=∣∣∣∣∣∣a11⋯an1⋯⋯a1nann∣∣∣∣∣∣DT=∣∣∣∣∣∣∣a11⋮a1n⋯⋯an1ann∣∣∣∣∣∣∣ -
行列式两行(列)互换，其值变号
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行列式某一行的公因子可以提到行列式符号的外面
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若行列式两行元素成比例，行列式为0
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∣ ⋯ a i 1 + b i 1 a i 2 + b i 2 ⋯ a i n + b i n ⋯ ∣ = ∣ ⋯ a i 1 a i 2 ⋯ a i n ⋯ ∣ + ∣ ⋯ b i 1 b i 2 ⋯ b i n ⋯ ∣ \left|\begin{array}{cccc} \ & \cdots& & \\ a_{i 1}+b_{i 1} & a_{i 2}+b_{i 2} & \cdots & a_{i n}+b_{i n} \\ & \cdots & \end{array}\right|=\left|\begin{array}{cccc} \ & \cdots& & \\ a_{i 1} & a_{i 2} & \cdots & a_{i n} \\ & \cdots & \end{array}\right|+\left|\begin{array}{cccc} \ & \cdots& & \\ b_{i 1} &b_{i 2} & \cdots &b_{i n} \\ & \cdots & \end{array}\right| ∣∣∣∣∣∣ ai1+bi1⋯ai2+bi2⋯⋯ain+bin∣∣∣∣∣∣=∣∣∣∣∣∣ ai1⋯ai2⋯⋯ain∣∣∣∣∣∣+∣∣∣∣∣∣ bi1⋯bi2⋯⋯bin∣∣∣∣∣∣
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D = r i + k r j D D \stackrel{r_{i}+k r_{j}}{=} D D=ri+krjD
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a i 1 A j 1 + a i 2 A j 2 + . . . + a i n A j n = { D i = j ( 展 开 定 理 ) 0 i ≠ j ( 推 论 ) a_{i1} A_{j 1}+a_{i 2} A_{j 2}+...+a_{i n} A_{j n}=\left\{\begin{array}{ll} D & i=j (展开定理)\\ 0 & i \neq j (推论) \end{array}\right. ai1Aj1+ai2Aj2+...+ainAjn={
D0i=j(展开定理)i=j(推论)
