矩阵和行列式知识点

矩阵和行列式复习

知识梳理

9.1矩阵的概念：

矩阵：像 ， ，

的矩形数字（或字母）阵列称为矩阵.通常用大写字母A 、

B 、C…表示

三个矩阵分别是2×1矩阵，2×2矩阵（二阶矩阵），2×3矩阵； ① 矩阵行的个数在前。

②矩阵相等：行数、列数相等，对应的元素也相等的两个矩阵，称为A ＝B 。 行向量、列向量

单位矩阵的定义：主对角线元素为1，其余元素均为0的矩阵 增广矩阵的含义及意义：在系数矩阵的右边添上线性方程组等号右边的值的矩阵。通过矩阵变换，解决多元一次方程的解。

9.2矩阵的运算 【矩阵加法】

不同阶的矩阵不可以相加；

记11

1221

22A A A A A =??

?

???，11

1221

22B B B B B =??

????

，那么

??

?

???++++=+22222121121211

11B A B A B A B A B A ， 【矩阵乘法】，

=11122122A B A B A B A B ??

????

；

??

?

???++++=22221221212211212212121121

1211

11B A B A B A B A B A B A B A B A AB

【矩阵的数乘】().ij kA Ak ka == 【矩阵变换】

相似变换的变换矩阵特点：k

等

轴对称变换的变换矩阵： 、 、

等

旋转变换的变换矩阵：

等

9.3二阶行列式

【行列式】行列式是由解线性方程组产生的一种算式； 行列式是若干数字组成的一个类似于矩阵的方阵，与矩阵不同的是，矩阵的表示是用中括号，而行列式则用线段。

行列式行数、列数一定相等；矩阵行数、列数不一定相等。 二阶行列式的值a d D ac bd b

c

==-

展开式ac -bd

【二元线性方程组】

对于二元一次方程组111222a x b y c a x b y c +=??+=?，通过加减消元法转化为方程组x

y D x D D y D ?=????=??

其中1

11

11

12

22222

,,x y a b c b a c D D D a b c b a c =

==

方程的解为

用行列式来讨论二元一次方程组解的情况。 （I ）0D ≠，方程组（*）有唯一解； （II ）0D =

○

1,x y D D 中至少有一个不为零，方程组（*）无解； ○

20x y D D ==，方程组（*）有无穷多解。 系数行列式11

2

2

a b D a b =

也为二元一次方程组解的判别式。

9.4三阶行列式

三阶行列式展开式及化简12

3

1

231232313121

2

3

a a a D

b b b a b

c a b c a b c c c c ==++321213132()

a b c a b c a b c -++（对角线法则）

三阶行列式的几何意义：直角坐标系中A 、B 、C 三点共线的充要条件(沪教P95)

【余子式】把三阶行列式中某个元素所在的行和列划去，将剩下的元素按原来位置关系组成的二阶行列式叫做该元素的余子式；添上符号（-1）i+j 后为代数余子式。

=a1A1+a2A2+a3A3

其中A1=, A2=-, A3=,分别为a1,a2,a3的代数余子式。

三阶行列式可以按照其任意一行或列展开成该行或列元素与其对应的代数余子式的乘积之和。

【三元线性方程组】

设三元一次方程组，其中x、y、z是未知数，通过加减消元化简为

,

，方程组（*）有唯一解；

巩固习题

1. （2018上海数学）行列式

41

25

的值为． 2. （2017上海数学）关于x 、y 的二元一次方程组

的系数行列式

D 为。

3. （2015上海数学）若线性方程组的增广矩阵为

解为 ，则

c 1-c 2=。 4. 函数1

sin cos 2)(-=

x x x f 的值域是.

5. (2018江苏数学)已知矩阵A=

，若点P 在矩阵 对应的变换作用下得到

点，求点P 的坐标．

7. 若行列式

中，元素4的代数余子式大于0，则x 满足的条件是________________ . 8. 行列式

所有可能的值中，最大的是_________ 。

9. 在n 行n 列矩阵

中，记位于第i 行第j 列的数为。当时，

_________ 。

10. 在数列

{}

n a 中，21

n n a =-，若一个7行12列的矩阵的第i 行第j 列的元素

,i j i j i j

a a a a a =?++，（12

,,7;12,,12i j ==

）则该矩阵元素能取到的不同数

值的个数为_________ 。

11. (2014上海数学)已知P 1(a 1,b 1)与P 2(a 2,b 2)是y=kx+1（k 为常数）上的两个

(,1,2,)ij a i j n =???9n =11223399a a a a +++???+=

不同点，则关于x和y的方程组的解的情况是（）。

A．无论k,P1,P2如何，总是无解 B. 无论k,P1,P2如何，总有唯一解

C．存在k,P1,P2，使之恰有两解D. 存在k,P1,P2，使之有无穷多解

12.当a为何值时，关于x,y,z的三元一次方程组有唯一解，并写出该条件下方程组的解。

参考答案

1.18

2.

3. 16

4.

5.（3，-1）

6.1

7.

8.27

9. 45

10.18

11.B 解析：由已知条件b1=ka1+1, b2=ka2+1 D==a1b2-a2b1=a1(ka2+1)-a2(ka1+1)=a1 -a20，∴有唯一解。

12.当a1时方程组的解为

赠送以下资料

英语万能作文(模板型）

Along with the advance of the society more and more problems are brought to our attention, one of which is that....

随着社会的不断发展，出现了越来越多的问题，其中之一便是____________。

As to whether it is a blessing or a curse, however, people take different attitudes.

然而，对于此类问题，人们持不同的看法。(Hold different attitudes 持不同的看;Come up with different attitudes 有不同的看法)

As society develops, people are attaching much importance to....

随着社会的发展，人们开始关注............

People are attaching more and more importance to the interview during job hunting

求职的过程中，人们慢慢意识到面试的重要性。

As to whether it is worthwhile ....., there is a long-running controversial debate. It is quite natural that people from different backgrounds may have divergent attitudes towards it.

关于是否值得___________的问题，一直以来争论不休。当然，不同的人对此可能持不同的观点。

In the process of modern urban development, we often find ourselves in a dilemma.

在都市的发展中，我们往往会陷入困境。

Recently the phenomenon has aroused wide concern, some people are in alarm that....

最近，这种现象引起了人们的广泛关注，有人开始担心______________。

The human race has entered a completely new stage in its history, with the increasingly rapid economic globalization and urbanization, more problems are brought to our attention.

人类进入了一个历史的崭新的阶段，经济全球化、都市化的速度

不断加快，随之给我们带来了很多问题。

...... plays such an important role that it undeniably becomes the biggest concern of the present world, there comes a question, is it a blessing or a curse?"

_______显得非常重要而成为当今世界所关注的最大的问题，这是无可厚非的。不过，问题是："我们该如何抉择?"

Now we are entering a new era, full of opportunities and challenges,

现在我们正在进入一个充满机会和挑战的新时代。

People from different backgrounds would put different interpretations on the same case.

不同行业的人对同一种问题的解释不尽相同。

The controversial issue is often brought into public focus. People from different backgrounds hold different attitudes towards the issue.

这中极具争议性的话题往往很受社会的关注。不同的人对此问题的看法也不尽相同。

When asked ..., some people think..... while some prefer...

说到______，有人认为________，而另一些人则认为__________。

Just as the saying goes: "so many people, so many minds". It is quite understandable that views on this issue vary from person to person.

俗话说，""。不同的人对此有不同的看法是可以理解的。

To this issue, different people come up with various attitudes.

对于这个问题，不同的人持不同的观点。

There is a good side and a bad side to everything, it goes without saying that...

万事万物都有其两面性，所以，勿庸置疑，____________。

When it comes to ..., most people believe that ..., but other people regard ...as ....

提到_________问题，很多人认为_________，不过，一些人则认为______是____.

When faced with...., quite a few people claim that ...., but other people think as...

提到_________问题，仅少数人认为________，但另一些人则认为_________。

There is a public controversy nowadays over the issue of .... There who criticize ...argue that ...., they believe that ...,but people who favor .., on the other hand, argue that...

目前，____问题争议较大。批判_____的人认为_____，他们认为_____，不过，另一方面，赞同_____的人则认为_______。

Some people are of the opinion that.. 有些人认为_________。

Many people claim that... 很多人认为_________。

A majority of 绝大多数

A large number of 很多人

Some people contend that ... has proved to bring many advantages (disadvantages)

有些人认为________有很多有利之处(不利之处)。

Those who argue for ... say that ...economic development of the cities.

觉得_____的人认为，______ 城市的经济发展。

Some people advocate that ....

有些人在坚持认为_________。

They hold that ... 他们认为_________。

People, who advocate that ..., have their sound reasons (grounds)

坚持认为______的人也有其说法(依据)。

Those who have already benefited from practicing it sing high praise of it. 那些从中受益的人对此大家褒奖。

Those who strongly approve of ... have cogent reasons for it.

强烈认同_______的人有很多原因。

Many people would claim that... 有人会认为___________。

People who support ... give some or all of the following

reasons.

那些支持_________观点的人列出了如下原因：________。

But others hold the view that ...但是，另外一些人则认为_______。

观点的用词：Attitude, opinion, 与其搭配的动词以及词组：Take, have, come up with，set forth, put forward等。

But on the other hand, there are also quite a few people who strongly advocate that..,.

不过，另一方面，也有少部分人坚持认为_______________。

But people who are ..., on the other hand , maintain that...

不过，另一方面，________的人认为__________。

However, there are a large number of people who hold a different view concerning this case.

然而，很多人对此有不同的看法。

问题用词：Issue, phenomenon，后接介词, on, over等。

However, some others argue that... 然而，另一些人则认为____。

However, there are also some others who contend that...

然而，也有人认为__________。

But other people set forth completely totally different argument concerning this case.

不过，对于此，另一些人则持完全不同的观点。

Some people examine this issue from another angle.

有的人用另一角度来看这一问题。

On the other hand, there are also many opponents who strongly ...

另一方面，也有很多反对的人，他们认为_____________。

According to my personality and fondness, I would prefer ... rather than...

根据我的个性以及兴趣，我选择_______而不会选择__________。

Personally, I side with the latter (former) opinion...

就我个人而言，我支持后者(前者)___________。

Personally, I am in favor of the former point of view.

就我个人而言，我较同意前一种看法。

To my point of view 我认为

To my mind, the advantages far overweigh the drawbacks(disadvantages, shortcomings)

我认为，优点胜过缺点。

For my part, I stand on side of the latter opinion that..

就我而言，我较赞同后一种观点________________。

As far as I am concerned, I am inclined to be on the side of the latter view.

在我看来，我较同意后一种观点。

After a thorough consideration, for my part, I am in favor

of the latter view that...

经过深思熟虑，我较支持后一种看法，亦即___________。

If asked to make a decision, I would prefer...

如果真的需要作出选择，我宁愿____________。

展现问题篇

问题的常用词：question, problem, issue

Recently, the issue of ...... has been brought into public focus.

近来，_______的问题引起了社会的广泛关注。

Now we are entering a brand new era full of opportunities and innovations, and great changes have taken place in people’s attitude towards some traditional practice.

现在我们进入了一个充满机遇和创新的崭新时代，很多人对某些传统的看法也发生了很大改变。

Recently the issue of whether or not ... has been in the limelight and has aroused wide concern in the public.

近来，是否____的问题已经非常明确而且引起了社会的广泛关注。

The issue whether it is good or not to .... has aroused a heated discussion all over the country.

______的利与弊已在全国范围内引起热烈的讨论。

At present, some people think ....while others claim ...Both sides have their merits.

目前，一些人认为_____而另一些人则认为_____。其实，两种观点都其可取之处。

People rarely reach an absolute consensus on such a controversial issue.

对于这种极具争议的话题，我们很难作出绝对的回答。People from different backgrounds would put different interpretations on the same case.

不同行业的人对同一种问题的解释不尽相同。

The controversial issue is often brought into public focus. People from different backgrounds hold different attitudes towards the issue.

这中极具争议性的话题往往很受社会的关注。不同的人对此问题的看法也不尽相同。

When asked ..., some people think..... while some prefer...

说到______，有人认为________，而另一些人则认为__________。

Just as the saying goes: "so many people, so many minds". It is quite understandable that views on this issue vary from person to person.

俗话说，""。不同的人对此有不同的看法是可以理解的。

To this issue, different people come up with various attitudes.

对于这个问题，不同的人持不同的观点。

There is a good side and a bad side to everything, it goes without saying that...

万事万物都有其两面性，所以，勿庸置疑，____________。

When it comes to ..., most people believe that ..., but other people regard ...as ....

提到____问题，很多人认为____，不过，一些人则认为____是____. When faced with...., quite a few people claim that ...., but other people think as...提到___问题，仅少数人认为___，但另一些人则认为____.