目錄:點我
思維導圖下載下傳:點我
一、重要概念
- 逆序數判定正負号
- 餘子式、代數餘子式與值無關
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克拉默法則:
x i = D i D x_i=\frac{D_i}{D} xi=DDi
推論1:若 ∣ A ∣ ≠ 0 |A|\ne0 ∣A∣=0,則方程隻有零解
推論2:若方程有非零解,則 ∣ A ∣ = 0 |A|=0 ∣A∣=0
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秩:
r ( A ) = 0 ⇔ A = O r(A)=0\Leftrightarrow A=O r(A)=0⇔A=O
A ≠ O ⇔ r ( A ) ≥ 1 A\ne O\Leftrightarrow r(A)\ge1 A=O⇔r(A)≥1
若 A A A為 m × n m\times n m×n矩陣,則:
r ( A ) ≤ min ( m , n ) r(A)\le \min{(m,n)} r(A)≤min(m,n)
二、重要公式
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公式:
A A ∗ = A ∗ A = ∣ A ∣ E AA^*=A^*A=|A|E AA∗=A∗A=∣A∣E
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四階以上副對角線符号判定:
∣ A ∣ = ( − 1 ) n ( n − 1 ) 2 a 1 n a 2 , n − 1 a n 1 |A|=(-1)^{\frac{n(n-1)}{2}}a_{1n}a_{2,n-1}a_{n1} ∣A∣=(−1)2n(n−1)a1na2,n−1an1
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拉普拉斯展開式:
∣ A ∗ O B ∣ = ∣ A O ∗ B ∣ = ∣ A ∣ ⋅ ∣ B ∣ ∣ O A B ∗ ∣ = ∣ ∗ A B O ∣ = ( − 1 ) m n ∣ A ∣ ⋅ ∣ B ∣ \begin{vmatrix} A & * \\ O & B \end{vmatrix} = \begin{vmatrix} A & O \\ {*} & B \end{vmatrix} =|A|\cdot|B| \\ \begin{vmatrix} O & A \\ B & * \end{vmatrix} = \begin{vmatrix} {*} & A \\ B & O \end{vmatrix} =(-1)^{mn}|A|\cdot|B| ∣∣∣∣AO∗B∣∣∣∣=∣∣∣∣A∗OB∣∣∣∣=∣A∣⋅∣B∣∣∣∣∣OBA∗∣∣∣∣=∣∣∣∣∗BAO∣∣∣∣=(−1)mn∣A∣⋅∣B∣
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範德蒙行列式:
∣ 1 1 … 1 x 1 x 2 … x n x 1 2 x 2 2 … x n 2 ⋮ ⋮ … ⋮ x 1 n x 2 n … x n n ∣ = ∏ 1 ≤ i < j ≤ n ( x i − x j ) \begin{vmatrix} 1 & 1 & \dots & 1 \\ x_1 & x_2 & \dots & x_n \\ x_1^2 & x_2^2 & \dots & x_n^2 \\ \vdots & \vdots & \dots & \vdots \\ x_1^n & x_2^n & \dots & x_n^n \\ \end{vmatrix} =\prod_{1\le i<j\le n}(x_i-x_j) ∣∣∣∣∣∣∣∣∣∣∣1x1x12⋮x1n1x2x22⋮x2n……………1xnxn2⋮xnn∣∣∣∣∣∣∣∣∣∣∣=1≤i<j≤n∏(xi−xj)
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行列式公式:
∣ A T ∣ = ∣ A ∣ |A^T|=|A| ∣AT∣=∣A∣
∣ k A ∣ = k n ∣ A ∣ |kA|=k^n|A| ∣kA∣=kn∣A∣
∣ A B ∣ = ∣ A ∣ ∣ B ∣ |AB|=|A||B| ∣AB∣=∣A∣∣B∣
∣ A ∗ ∣ = ∣ A ∣ n − 1 |A^*|=|A|^{n-1} ∣A∗∣=∣A∣n−1
∣ A − 1 ∣ = ∣ A ∣ − 1 |A^{-1}|=|A|^{-1} ∣A−1∣=∣A∣−1
∣ A ∣ = ∏ i = 1 n λ i |A|=\prod_{i=1}^{n}\lambda_i ∣A∣=i=1∏nλi
若A與B相似,則:
∣ A ∣ = ∣ B ∣ |A|=|B| ∣A∣=∣B∣