问题 多项选择题

设二次型f(x1,x2,x3)=XTAX在正交变换X=PY下的标准形为

,且P的第三列为


(Ⅰ) 求A;
(Ⅱ) 证明A+E为正定矩阵.

答案

参考答案:因为二次型f(xA,xB,xC)=XTAX在正交变换X=PY下的标准形为[*],所以A的特征值为λAB=A,λC=0.P的第三列为[*],所以λC=0对应的线性无关的特征向量为[*].因为A为实对称矩阵,所以A的不同特征值对应的特征向量正交,令λAB=A对应的特征向量为[*].
由xA+xC=0的λAB=A对应的线性无关的特征向量为[*]
[*]
(Ⅱ) 因为[*]是实对称阵,且A的特征值为λAB=A,λC=0,所以A+E的特征值为λAB一B,λC=A,因为其特征值都大于零,所以A+E为正定矩阵.

解析:

[分析]: 利用实对称阵的性质导出A;由实对称阵的特征值全大于零证明A+E正定.

阅读理解

Want to stay away from colds? Put on a happy face.

Compared to unhappy people, those who are cheerful and relaxed are less likely to suffer from colds, according to a new study. It’s possible that being happy helps the body fight illnesses, say the researchers from New York University.

“It seems that positive feelings may reduce the danger of illness,” said the study’s chief researcher Sheldon Cohen.

In an earlier study, Cohen found that people who were cheerful and lively caught coughs and colds less often. People who showed feelings were also less likely to tell their doctors that they felt ill.

In this study, Cohen’s interviewed 193 adults every day for two weeks. During the interviews, the people told researchers about were given colds by doctors and had to stay alone in a room for six days.

The results showed that everyone in the study was equally likely to get ill. Buy for people who said they felt happy during the research period, their illness are less serious and lasted for a shorter time.

Cohen believes that when people experience positive feelings, their body may produce a chemical that helps fight illness and disease. So if you are worried abut your health, look on the bright side more often.

小题1: Which of the following was NOT a part of the study?

A.People talked about their feelings every day.

B.People were kept alone for six days.

C.People were given colds by doctors.

D.People were made to feel unhappy.小题2: What did the study find?

A.People who felt happy never got ill.

B.People’s feelings didn’t influence their health.

C.People with good feelings became ill more easily.

D.People with positive feelings had less serious illnesses..小题3: According to Cohen, which of the following may help fight illness?

A.Eating.

B.Crying.

C.Laughing.

D.Sleeping.小题4:This passage is a/an          .

A.advertisement

B.newspaper report

C.story

D.scientist’s diary小题5:What is the best title for this passage?

A.Smiles can fight colds

B.Cause of colds found

C.The danger of colds

D.How people get sick小题6:

单项选择题