行列式 Determinant
定义 Definition
行列式: 行列式是方阵的一个标量值，用来描述方阵的某些性质，如可逆性。对于一个 矩阵 ，其行列式记作 或 Determinant: The determinant is a scalar value that describes certain properties of a square matrix, such as invertibility. For an matrix , its determinant is denoted as or
计算 Calculation
-
矩阵的行列式 Determinant of a Matrix:
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \
a_{21} & a_{22} & a_{23} \
a_{31} & a_{32} & a_{33}
\end{pmatrix}
|A| = a_{11} \begin{vmatrix}
a_{22} & a_{23} \
a_{32} & a_{33}
\end{vmatrix} - a_{12} \begin{vmatrix}
a_{21} & a_{23} \
a_{31} & a_{33}
\end{vmatrix} + a_{13} \begin{vmatrix}
a_{21} & a_{22} \
a_{31} & a_{32}
\end{vmatrix}
3. **高阶行列式 Higher-Order Determinants**: 可以通过递归使用 $2 \times 2$ 和 $3 \times 3$ 矩阵的行列式公式来计算更高阶的行列式 Higher-order determinants can be computed recursively using the $2 \times 2$ and $3 \times 3$ determinant formulas ## 拉普拉斯展开 Laplace Expansion ### 定义 Definition **拉普拉斯展开**: 行列式可以通过任意一行或一列的元素及其余子式来展开 **Laplace Expansion**: A determinant can be expanded along any row or column using its elements and their minors ### 计算 Calculation 对于一个 $n \times n$ 矩阵 $A$，其行列式 $|A|$ 可以沿第 $i$ 行展开为: For an $n \times n$ matrix $A$, its determinant $|A|$ can be expanded along the $i$-th row as:|A| = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}
也可以沿第 $j$ 列展开: It can also be expanded along the $j$-th column:|A| = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}
## 余子式 Minor ### 定义 Definition **余子式**: 对于一个 $n \times n$ 矩阵 $A$，其余子式 $M_{ij}$ 是通过删除 $A$ 的第 $i$ 行和第 $j$ 列得到的 $(n-1) \times (n-1)$ 子矩阵的行列式 **Minor**: For an $n \times n$ matrix $A$, the minor $M_{ij}$ is the determinant of the $(n-1) \times (n-1)$ submatrix obtained by deleting the $i$-th row and $j$-th column from $A$ ### 计算 Calculation 设 $A$ 为一个 $3 \times 3$ 矩阵: Let $A$ be a $3 \times 3 matrix:A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \
a_{21} & a_{22} & a_{23} \
a_{31} & a_{32} & a_{33}
\end{pmatrix}
余子式 $M_{11}$ 通过删除第 $1$ 行和第 $1$ 列得到: The minor $M_{11}$ is obtained by deleting the 1st row and 1st column:M_{11} = \begin{vmatrix}
a_{22} & a_{23} \
a_{32} & a_{33}
\end{vmatrix} = a_{22}a_{33} - a_{23}a_{32}
同理，其他余子式可以类似计算 Similarly, other minors can be calculated similarly ## 代数余子式 Cofactor ### 定义 Definition **代数余子式**: 对于一个 $n \times n$ 矩阵 $A$，其代数余子式 $C_{ij}$ 是余子式 $M_{ij}$ 乘以 $(-1)^{i+j}$ **Cofactor**: For an $n \times n$ matrix $A$, its cofactor $C_{ij}$ is the minor $M_{ij}$ multiplied by $(-1)^{i+j}$ ### 计算 Calculation 代数余子式 $C_{ij}$ 计算如下: The cofactor $C_{ij}$ is calculated as follows:C_{ij} = (-1)^{i+j} M_{ij}
## 伴随矩阵 Adjugated Matrix ### 定义 Definition **伴随矩阵**: 对于一个 $n \times n$ 矩阵 $A$，其伴随矩阵 $\text{adj}(A)$ 是由 $A$ 的余子式组成的转置矩阵，即 **Adjugated Matrix**: For an $n \times n$ matrix $A$, its adjugated matrix $\text{adj}(A)$ is the transpose of the matrix of its cofactors:\text{adj}(A) = C
### 计算 Calculation 对于一个 $3 \times 3$ 矩阵 $A$，其伴随矩阵 $\text{adj}(A)$ 为: For a $3 \times 3$ matrix $A$, its adjugate matrix $\text{adj}(A)$ is:\text{adj}(A) = \begin{pmatrix}
C_{11} & C_{21} & C_{31} \
C_{12} & C_{22} & C_{32} \
C_{13} & C_{23} & C_{33}
\end{pmatrix}
其中 $C_{ij} = (-1)^{i+j} M_{ij}$ 是 $A$ 的余子式 $M_{ij}$ 的代数余子式 where $C_{ij} = (-1)^{i+j} M_{ij}$ is the cofactor of the minor $M_{ij}$ of $A$ ## 矩阵的逆 Inverse of a Matrix ### 计算 Calculation 若矩阵 $A$ 是可逆的，则其逆矩阵 $A^{-1}$ 为: If a matrix $A$ is invertible, its inverse $A^{-1}$ is given by:A^{-1} = \frac{1}{|A|} \text{adj}(A)
## 示例 Example ### 计算 $3 \times 3$ 矩阵的行列式和伴随矩阵 Calculate the Determinant and Adjugated Matrix of a $3 \times 3$ Matrix 设 $A$ 为一个 $3 \times 3$ 矩阵: Let $A$ be a $3 \times 3$ matrix:A = \begin{pmatrix}
1 & 2 & 3 \
0 & 4 & 5 \
1 & 0 & 6
\end{pmatrix}
|A| = 1 \begin{vmatrix}
4 & 5 \
0 & 6
\end{vmatrix} - 2 \begin{vmatrix}
0 & 5 \
1 & 6
\end{vmatrix} + 3 \begin{vmatrix}
0 & 4 \
1 & 0
\end{vmatrix}
= 1(24) - 2(-5) + 3(-4) = 24 + 10 - 12 = 22
M_{11} = \begin{vmatrix}
4 & 5 \
0 & 6
\end{vmatrix} = 24, \quad M_{12} = \begin{vmatrix}
0 & 5 \
1 & 6
\end{vmatrix} = -5, \quad M_{13} = \begin{vmatrix}
0 & 4 \
1 & 0
\end{vmatrix} = -4
M_{21} = \begin{vmatrix}
2 & 3 \
0 & 6
\end{vmatrix} = 12, \quad M_{22} = \begin{vmatrix}
1 & 3 \
1 & 6
\end{vmatrix} = 3, \quad M_{23} = \begin{vmatrix}
1 & 2 \
1 & 0
\end{vmatrix} = -2
M_{31} = \begin{vmatrix}
2 & 3 \
4 & 5
\end{vmatrix} = -2, \quad M_{32} = \begin{vmatrix}
1 & 3 \
0 & 5
\end{vmatrix} = 5, \quad M_{33} = \begin{vmatrix}
1 & 2 \
0 & 4
\end{vmatrix} = 4
C_{11} = 24, \quad C_{12} = 5, \quad C_{13} = -4
C_{21} = -12, \quad C_{22} = 3, \quad C_{23} = 2
C_{31} = -2, \quad C_{32} = -5, \quad C_{33} = 4
\text{adj}(A) = \begin{pmatrix}
24 & -12 & -2 \
5 & 3 & -5 \
-4 & 2 & 4
\end{pmatrix}^T = \begin{pmatrix}
24 & 5 & -4 \
-12 & 3 & 2 \
-2 & -5 & 4
\end{pmatrix}
A^{-1} = \frac{1}{|A|} \text{adj}(A) = \frac{1}{22} \begin{pmatrix}
24 & 5 & -4 \
-12 & 3 & 2 \
-2 & -5 & 4
\end{pmatrix}