# 线性代数作业代写linear algebra代考|PROBLEMS

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|diagonal matrix

1. Let $A=\left[\begin{array}{rr}1 & 4 \ -3 & 1\end{array}\right]$. Prove that $A$ is non-singular, find $A^{-1}$ and express $A$ as a product of elementary row matrices. [Answer: $A^{-1}=\left[\begin{array}{rr}\frac{1}{13} & -\frac{4}{13} \ \frac{3}{13} & \frac{1}{13}\end{array}\right]$, $A=E_{21}(-3) E_{2}(13) E_{12}(4)$ is one such decomposition.]
2. A square matrix $D=\left[d_{i j}\right]$ is called diagonal if $d_{i j}=0$ for $i \neq j$. (That is the off-diagonal elements are zero.) Prove that pre-multiplication of a matrix $A$ by a diagonal matrix $D$ results in matrix $D A$ whose rows are the rows of $A$ multiplied by the respective diagonal elements of $D$. State and prove a similar result for post-multiplication by a diagonal matrix.

Let $\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right)$ denote the diagonal matrix whose diagonal elements $d_{i i}$ are $a_{1}, \ldots, a_{n}$, respectively. Show that $$\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right) \operatorname{diag}\left(b_{1}, \ldots, b_{n}\right)=\operatorname{diag}\left(a_{1} b_{1}, \ldots, a_{n} b_{n}\right)$$ and deduce that if $a_{1} \ldots a_{n} \neq 0$, then $\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right)$ is non-singular and $$\left(\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right)\right)^{-1}=\operatorname{diag}\left(a_{1}^{-1}, \ldots, a_{n}^{-1}\right)$$ Also prove that $\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right)$ is singular if $a_{i}=0$ for some $i$.

1. Let $A=\left[\begin{array}{lll}0 & 0 & 2 \ 1 & 2 & 6 \ 3 & 7 & 9\end{array}\right]$. Prove that $A$ is non-singular, find $A^{-1}$ and express $A$ as a product of elementary row matrices. [Answers: $A^{-1}=\left[\begin{array}{rrr}-12 & 7 & -2 \ \frac{9}{2} & -3 & 1 \ \frac{1}{2} & 0 & 0\end{array}\right]$, $A=E_{12} E_{31}(3) E_{23} E_{3}(2) E_{12}(2) E_{13}(24) E_{23}(-9)$ is one such decomposition.]

## 线性代数作业代写linear algebra代考|singular

1. Find the rational number $k$ for which the matrix $A=\left[\begin{array}{rrr}1 & 2 & k \ 3 & -1 & 1 \ 5 & 3 & -5\end{array}\right]$ is singular. [Answer: $k=-3$.]
2. Prove that $A=\left[\begin{array}{rr}1 & 2 \ -2 & -4\end{array}\right]$ is singular and find a non-singular matrix $P$ such that $P A$ has last row zero.
3. If $A=\left[\begin{array}{rr}1 & 4 \ -3 & 1\end{array}\right]$, verify that $A^{2}-2 A+13 I_{2}=0$ and deduce that $A^{-1}=-\frac{1}{13}\left(A-2 I_{2}\right)$.
4. Let $A=\left[\begin{array}{rrr}1 & 1 & -1 \ 0 & 0 & 1 \ 2 & 1 & 2\end{array}\right]$. (i) Verify that $A^{3}=3 A^{2}-3 A+I_{3}$. (ii) Express $A^{4}$ in terms of $A^{2}, A$ and $I_{3}$ and hence calculate $A^{4}$ explicitly. (iii) Use (i) to prove that $A$ is non-singular and find $A^{-1}$ explicitly. [Answers: (ii) $A^{4}=6 A^{2}-8 A+3 I_{3}=\left[\begin{array}{rrr}-11 & -8 & -4 \ 12 & 9 & 4 \ 20 & 16 & 5\end{array}\right]$; (iii) $A^{-1}=A^{2}-3 A+3 I_{3}=\left[\begin{array}{rrr}-1 & -3 & 1 \ 2 & 4 & -1 \ 0 & 1 & 0\end{array}\right]$.]
5. (i) Let $B$ be an $n \times n$ matrix such that $B^{3}=0$. If $A=I_{n}-B$, prove that $A$ is non-singular and $A^{-1}=I_{n}+B+B^{2}$. Show that the system of linear equations $A X=b$ has the solution $$X=b+B b+B^{2} b .$$ (ii) If $B=\left[\begin{array}{ccc}0 & r & s \ 0 & 0 & t \ 0 & 0 & 0\end{array}\right]$, verify that $B^{3}=0$ and use (i) to determine $\left(I_{3}-B\right)^{-1}$ explicitly. [Answer: $\left[\begin{array}{rrr}1 & r & s+r t \ 0 & 1 & t \ 0 & 0 & 1\end{array}\right]$.]

## 线性代数作业代写linear algebra代考|diagonal matrix

1. 让一种=[14 −31]. 证明一种是非奇异的，找到一种−1并表达一种作为基本行矩阵的乘积。 [回答：一种−1=[113−413 313113], 一种=和21(−3)和2(13)和12(4)就是这样一种分解。]
2. 方阵D=[d一世j]被称为对角线如果d一世j=0为了一世≠j. （即非对角元素为零。）证明矩阵的预乘一种由对角矩阵D结果矩阵D一种其行是一种乘以相应的对角线元素D. 陈述并证明对角矩阵后乘的类似结果。

1. 让一种=[002 126 379]. 证明一种是非奇异的，找到一种−1并表达一种作为基本行矩阵的乘积。 [答案：一种−1=[−127−2 92−31 1200], 一种=和12和31(3)和23和3(2)和12(2)和13(24)和23(−9)就是这样一种分解。]

## 线性代数作业代写linear algebra代考|singular

1. 找到有理数到矩阵一种=[12到 3−11 53−5]是单数。[回答：到=−3.]
2. 证明一种=[12 −2−4]是奇异的并找到一个非奇异矩阵磷这样磷一种最后一行为零。
3. 如果一种=[14 −31], 验证一种2−2一种+13一世2=0并推断出一种−1=−113(一种−2一世2).
4. 让一种=[11−1 001 212]. (i) 核实一种3=3一种2−3一种+一世3. (ii) 快递一种4按照一种2,一种和一世3并因此计算一种4明确地。 (iii) 使用 (i) 证明一种是非奇异的并找到一种−1明确地。 [答案：（二）一种4=6一种2−8一种+3一世3=[−11−8−4 1294 20165]; ㈢一种−1=一种2−3一种+3一世3=[−1−31 24−1 010].]
5. （我让乙豆角，扁豆n×n矩阵使得乙3=0. 如果一种=一世n−乙， 证明一种是非奇异的并且一种−1=一世n+乙+乙2. 证明线性方程组一种X=b有解决办法 X=b+乙b+乙2b. (ii) 如果乙=[0rs 00吨 000], 验证乙3=0并使用 (i) 确定(一世3−乙)−1明确地。 [答案：[1rs+r吨 01吨 001].]

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions