# 线性代数作业代写linear algebra代考| Non–singular matrices

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• 矩阵论
• 优化理论
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• 逼近论

## 线性代数作业代写linear algebra代考|Non–singular matrices

A square matrix $A \in M_{n \times n}(F)$ is called non-singular or invertible if there exists a matrix $B \in M_{n \times n}(F)$ such that
$$A B=I_{n}=B A$$
Any matrix $B$ with the above property is called an inverse of $A$. If $A$ does not have an inverse, $A$ is called singular.

THEOREM 2.5.1 (Inverses are unique)
If $A$ has inverses $B$ and $C$, then $B=C$.
Proof. Let $B$ and $C$ be inverses of $A$. Then $A B=I_{n}=B A$ and $A C=$ $I_{n}=C A$. Then $B(A C)=B I_{n}=B$ and $(B A) C=I_{n} C=C$. Hence because $B(A C)=(B A) C$, we deduce that $B=C$.
REMARK 2.5.1 If $A$ has an inverse, it is denoted by $A^{-1}$. So
$$A A^{-1}=I_{n}=A^{-1} A$$
Also if $A$ is non-singular, it follows that $A^{-1}$ is also non-singular and
$$\left(A^{-1}\right)^{-1}=A$$
THEOREM 2.5.2 If $A$ and $B$ are non-singular matrices of the same size, then so is $A B$. Moreover
$$(A B)^{-1}=B^{-1} A^{-1}$$
Proof.
$(A B)$
Similarly $$\left(B^{-1} A^{-1}\right)(A B)=I_{n} .$$
REMARK 2.5.2 The above result generalizes to a product of $m$ nonsingular matrices: If $A_{1}, \ldots, A_{m}$ are non-singular $n \times n$ matrices, then the product $A_{1} \ldots A_{m}$ is also non-singular. Moreover
$$\left(A_{1} \ldots A_{m}\right)^{-1}=A_{m}^{-1} \ldots A_{1}^{-1} .$$
(Thus the inverse of the product equals the product of the inverses in the reverse order.)

The system
\begin{aligned} &a x+b y=e \ &c x+d y=f \end{aligned}
has a unique solution if $\Delta=\left|\begin{array}{ll}a & b \ c & d\end{array}\right| \neq 0$, namely
$$x=\frac{\Delta_{1}}{\Delta}, \quad y=\frac{\Delta_{2}}{\Delta},$$
where
$$\Delta_{1}=\left|\begin{array}{cc} e & b \ f & d \end{array}\right| \quad \text { and } \quad \Delta_{2}=\left|\begin{array}{cc} a & e \ c & f \end{array}\right| .$$

## 线性代数作业代写linear algebra代考|Non–singular matrices

(一种−1)−1=一种

(一种乙)−1=乙−1一种−1

(一种乙)

(一种1…一种米)−1=一种米−1…一种1−1.
（因此，乘积的倒数等于倒序的倒数的乘积。）

X=Δ1Δ,和=Δ2Δ,

Δ1=|和b Fd| 和 Δ2=|一种和 CF|.

# 计量经济学代写

## 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