Posted on Categories:Linear algebra, 数学代写, 线性代数

# 数学代写|线性代数代写Linear algebra代考|The Cofactor Method

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|线性代数代写Linear algebra代考|The Cofactor Method

Recall that for $\mathbf{A} \in \mathbb{R}^{n \times n}$ we defined
$$\operatorname{det} \mathbf{A}=a_{j 1} C_{j 1}+a_{j 2} C_{j 2}+\cdots+a_{j n} C_{j n}$$
where $C_{j k}=(-1)^{j+k} \operatorname{det} \mathbf{A}{j k}$ is called the $(j, k)$-Cofactor of $\mathbf{A}$ and $$\mathbf{a}_j=\left[\begin{array}{llll} a{j 1} & a_{j 2} & \cdots & a_{j n} \end{array}\right]$$
is the $j$ th row of $\mathbf{A}$. If $\mathbf{c}j=\left[\begin{array}{llll}C{j 1} & C_{j 2} & \cdots & C_{j n}\end{array}\right]$ then
$$\operatorname{det} \mathbf{A}=\left[\begin{array}{llll} a_{j 1} & a_{j 2} & \cdots & a_{j n} \end{array}\right]\left[\begin{array}{c} C_{j 1} \ C_{j 2} \ \vdots \ C_{j n} \end{array}\right]=\mathbf{a}j \cdot \mathbf{c}_j^T$$ Suppose that $\mathbf{B}$ is the matrix obtained from $\mathbf{A}$ by replacing row $\mathbf{a}_j$ with a distinct row $\mathbf{a}_k$. To compute det $\mathbf{B}$ expand along its $j$ th row $\mathbf{b}_j=\mathbf{a}{\mathbf{k}}$ :
$$\operatorname{det} \mathbf{B}=\mathbf{a}_k \cdot \mathbf{c}_j^T=0$$
The Cofactor Method is an alternative method to find the inverse of an invertible matrix. Recall that for any matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$, if we expand along the $j$ th row then
$$\operatorname{det} \mathbf{A}=\mathbf{a}_j \cdot \mathbf{c}_j^T$$
On the other hand, if $j \neq k$ then
$$\mathbf{a}_j \cdot \mathbf{c}_k^T=0$$
In summary,
$$\mathbf{a}_j \cdot \mathbf{c}_k^T= \begin{cases}\operatorname{det} \mathbf{A}, & \text { if } j=k \ 0, & \text { if } j \neq k\end{cases}$$

## 数学代写|线性代数代写Linear algebra代考|Volumes

The volume of the parallelepiped determined by the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ is
$$\operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right)=\operatorname{abs}\left(\mathbf{v}_1^T\left(\mathbf{v}_2 \times \mathbf{v}_3\right)\right)=\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{lll} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{array}\right]\right)$$
where $\operatorname{abs}(x)$ denotes the absolute value of the number $x$. Let $\mathbf{A}$ be an invertible matrix and let $\mathbf{w}_1=\mathbf{A} \mathbf{v}_1, \mathbf{w}_2=\mathbf{A} \mathbf{v}_2, \mathbf{w}_3=\mathbf{A} \mathbf{v}_3$. How are $\operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_2\right)$ and $\operatorname{Vol}\left(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_2\right)$ related? Compute:
\begin{aligned} \operatorname{Vol}\left(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3\right) & =\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{lll} \mathbf{w}_1 & \mathbf{w}_2 & \mathbf{w}_3 \end{array}\right]\right) \ & =\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{lll} \mathbf{A v}_1 & \mathbf{A v}_2 & \mathbf{A} \mathbf{v}_3 \end{array}\right]\right) \ & =\operatorname{abs}\left(\operatorname{det}\left(\mathbf{A}\left[\begin{array}{lll} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{array}\right]\right)\right) \ & =\operatorname{abs}\left(\operatorname{det} \mathbf{A} \cdot \operatorname{det}\left[\begin{array}{lll} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{array}\right]\right) \ & =\operatorname{abs}(\operatorname{det} \mathbf{A}) \cdot \operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right) . \end{aligned}
Therefore, the number abs(det $\mathbf{A})$ is the factor by which volume is changed under the linear transformation with matrix A. In summary:

Theorem 13.4: Suppose that $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are vectors in $\mathbb{R}^3$ that determine a parallelepiped of non-zero volume. Let $\mathbf{A}$ be the matrix of a linear transformation and let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3$ be the images of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ under $\mathbf{A}$, respectively. Then
$$\operatorname{Vol}\left(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3\right)=\operatorname{abs}(\operatorname{det} \mathbf{A}) \cdot \operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right)$$

## 数学代写|线性代数代写Linear algebra代考|The Cofactor Method

$$\operatorname{det} \mathbf{A}=a_{j 1} C_{j 1}+a_{j 2} C_{j 2}+\cdots+a_{j n} C_{j n}$$

$$\mathbf{a}j=\left[\begin{array}{llll} a j 1 & a{j 2} & \cdots & a_{j n} \end{array}\right]$$

$$\operatorname{det} \mathbf{A}=\left[\begin{array}{llll} a_{j 1} & a_{j 2} & \cdots & a_{j n} \end{array}\right]\left[\begin{array}{ll} C_{j 1} C_{j 2} \vdots C_{j n} \end{array}\right]=\mathbf{a} j \cdot \mathbf{c}_j^T$$

$$\operatorname{det} \mathbf{B}=\mathbf{a}_k \cdot \mathbf{c}_j^T=0$$

$$\operatorname{det} \mathbf{A}=\mathbf{a}_j \cdot \mathbf{c}_j^T$$

$$\mathbf{a}_j \cdot \mathbf{c}_k^T=0$$

$$\mathbf{a}_j \cdot \mathbf{c}_k^T={\operatorname{det} \mathbf{A}, \quad \text { if } j=k 0, \quad \text { if } j \neq k$$

## 数学代写|线性代数代写Linear algebra代考|Volumes

$$\operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right)=\operatorname{abs}\left(\mathbf{v}_1^T\left(\mathbf{v}_2 \times \mathbf{v}_3\right)\right)=\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{ccc} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{array}\right]\right)$$

$$\operatorname{Vol}\left(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3\right)=\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{lll} \mathbf{w}_1 & \mathbf{w}_2 & \mathbf{w}_3 \end{array}\right]\right) \quad=\operatorname{abs}\left(\operatorname{det}\left[\begin{array}{lll} \mathbf{A} \mathbf{v}_1 & \mathbf{A v}_2 & \mathbf{A} \mathbf{v}_3 \end{array}\right]\right)=\operatorname{abs}\left(\operatorname{det}\left(\mathbf{A}\left[\mathbf{v}_1 \quad \mathbf{v}_2 \quad \mathbf{v}_3\right]\right)\right)$$

$$\operatorname{Vol}\left(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3\right)=\operatorname{abs}(\operatorname{det} \mathbf{A}) \cdot \operatorname{Vol}\left(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。