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# 数学代写| Determinants 离散数学代写

## 数学代写| Determinants 代考

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## 离散数学代写

The determinant is a function defined on square matrices and its value is a scalar. A key property of determinants is that a matrix is invertible if and only if its determinant is non-zero. The determinant of a $2 \times 2$ matrix is given by
$$\left|\begin{array}{ll} a & b \ c & d \end{array}\right|=a d-b c$$
The determinant of a $3 \times 3$ matrix is given by
$$\left|\begin{array}{llc} a & b & c \ d & e & f \ g & h & i \end{array}\right|=a e i+b f g+c d h-a f h-b d i-c e g$$
Cofactors
Let $A$ be an $n \times n$ matrix. For $1 \leq i, j \leq n$, the $(i, j)$ minor of $A$ is defined to be the $(n-1) \times(n-1)$ matrix obtained by deleting the $i$-th row and $j$-th column of $A$ (Fig. 8.5).
140
8 Matrix Theory
Fig. $8.5$ Determining the $(i, j)$ minor of $\mathrm{A}$
$$\left(\begin{array}{cccccc} a_{11} & a_{12} & \ldots . & a_{1 j} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 j} & \ldots & a_{2 n} \ a_{31} & a_{32} & \ldots . & a_{3 j} & \ldots . & a_{3 n} \ \ldots & \ldots . & \ldots & \ldots . & \ldots . & \ldots . \ a_{i 1} & a_{i 2} & \ldots . & a_{i j} & \ldots . & a_{i n} \ \ldots & \ldots . & \ldots . & \ldots . & \ldots . & \ldots . \ a_{n 1} & a_{n 2} & \ldots . & a_{n j} & \ldots . & a_{n n} \end{array}\right)$$
$i, j$ minor of $A$
The shaded row is the $i$ th row, and the shaded column is the $j$ th column. These both are deleted from A to form the $(i, j)$ minor of A, and this is a $(n-1) \times(n-1)$ matrix.

The $(i, j)$ cofactor of $A$ is defined to be $(-1)^{i+j}$ times the determinant of the $(i, j)$ minor. The $(i, j)$ cofactor of $A$ is denoted by $K_{i},(A)$.

The cofactor matrix Cof A is formed in this way where the $(i, j)$ th element in the cofactor matrix is the $(i, j)$ cofactor of A.
Definition of Determinant
The determinant of a matrix is defined as
$$\operatorname{det} \mathrm{A}=\sum_{j=1}^{n} A_{i j} K_{i j} .$$
Another words the determinant of $\mathrm{A}$ is determined by taking any row of $A$ and multiplying each element by the corresponding cofactor and adding the results. The determinant of the product of two matrices is the product of their determinants.
$$\operatorname{det}(A B)=\operatorname{det} A \times \operatorname{det} B$$
Definition
The adjugate of $A$ is the $n \times n$ matrix $A d j(A)$ whose $(i, j)$ entry is the $(j, i)$ cofactor $K_{j i}(A)$ of $A$. That is, the adjugate of $\mathrm{A}$ is the transpose of the cofactor matrix of $\mathrm{A}$. Inverse of $\mathbf{A}$

The inverse of $\mathrm{A}$ is determined from the determinant of $\mathrm{A}$ and the adjugate of $\mathrm{A}$. That is,
$$\mathrm{A}^{-1}=\frac{1}{\operatorname{det} \mathrm{A}} \operatorname{Adj} A=\frac{1}{\operatorname{det} \mathrm{A}}(\operatorname{Cof} \mathrm{A})^{\mathrm{T}}$$

$$\left|\begin{数组}{ll} a & b \ 开发 \end{数组}\right|=a d-b c$$
$3 \times 3$ 矩阵的行列式由下式给出
$$\left|\begin{数组}{llc} a & b & c \ d & e & f \ g&h&我 \end{数组}\right|=a e i+b f g+c d h-a f h-b d i-c e g$$

140
8 矩阵理论

$$\left(\begin{数组}{cccccc} a_{11} & a_{12} & \ldots 。 & a_{1 j} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 j} & \ldots & a_{2 n} \ a_{31} & a_{32} & \ldots 。 & a_{3 j} & \ldots 。 & a_{3 n} \ \ldots & \ldots 。 & \ldots & \ldots 。 & \ldots 。 & \ldots 。 \ a_{i 1} & a_{i 2} & \ldots 。 & a_{i j} & \ldots 。 & a_{i n} \ \ldots & \ldots 。 & \ldots 。 & \ldots 。 & \ldots 。 & \ldots 。 \ a_{n 1} & a_{n 2} & \ldots 。 & a_{n j} & \ldots 。 & a_{n n} \end{数组}\右）$$
$a$ 的 $i, j$ 小调

$A$ 的 $(i, j)$ 辅因子定义为 $(-1)^{i+j}$ 乘以 $(i, j)$ 小调的行列式。 $A$ 的 $(i, j)$ 辅因子用 $K_{i},(A)$ 表示。

$$\operatorname{det} \mathrm{A}=\sum_{j=1}^{n} A_{i j} K_{i j} 。$$

$$\operatorname{det}(A B)=\operatorname{det} A \times \operatorname{det} B$$

$A$ 的对数是 $n \times n$ 矩阵 $A dj(A)$，它的 $(i, j)$ 条目是 $(j, i)$ 辅因子 $K_{ji}(A)$澳元。也就是说，$\mathrm{A}$ 的对数是 $\mathrm{A}$ 的辅因子矩阵的转置。 $\mathbf{A}$ 的逆

$\mathrm{A}$ 的逆由 $\mathrm{A}$ 的行列式和 $\mathrm{A}$ 的并量决定。那是，
$$\mathrm{A}^{-1}=\frac{1}{\operatorname{det} \mathrm{A}} \operatorname{Adj} A=\frac{1}{\operatorname{det} \mathrm{A }}(\operatorname{Cof} \mathrm{A})^{\mathrm{T}}$$

## 密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码