问题 问答题

设A为三阶矩阵,有三个不同特征值λ1,λ2,λ3,对应的特征向量依次为α1,α2,α3,令β=α123

证明:β不是A的特征向量

答案

参考答案:证一 假设β为A的特征向量,则存在λ0,使Aβ=λ0β,即
得(λ101+(λ202+(λ303=0.由α1,α2,α3线性无关知从而有λ123,这与已知条件矛盾,因此β不是A的特征向量.
证二 因α1,α2,α3是属于不同特征值的特征向量,故α123必不是A的

解析: 可用反证法证之

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Dad is a liar (说谎者), definitely.

He never tells the truth about anything   36  , or anything that he thinks is bad to me, even though I   37  that I don’t mind hearing these things.

He is the kindest man I have ever met , who never   38  others, as well as the    39 man, for he’s always hard on himself. Every time I   40  home from college and asked Dad how his   41 was getting on, he would always reply. “Oh, couldn’t be   42  !”when I asked my mum the same question on the phone ,   43  , she honestly told me every   44 with Dad’s business. I didn’t blame Dad for his   45  . I felt a deep sympathy for him.

Dad is a miser (吝啬鬼), undoubtedly..

I hardly see Dad wear   46  clothes. In fact , his closet is half empty. Even in this half, two-thirds is occupied by Mum’s clothes and the other   47    belongs to him . I urged him to buy some new clothes ,   48  the simply shook his head, “The old clothes are still good enough.” Were they? I saw   49  in them.

It   50  my heart up when I saw had Dad  51  terribly with his hand covering his mouth. When the pains became unbearable, he   52 took some pills.

So it surprised   53   when the day came that Dad got sick . He was lying in bed. And all the family gathered around him. I knelt (跪)by his bedside , tears filling my eyes.

Dear Dad, you’ve been   54  yourself too hard, which you should not have. I know I might as well   55  a river to flow backward as hope to talk you out of working so hard. But I still want to say . “Dad , take better care of yourself!”

36.A.interesting                     B.good               C.bad                 D.valuable

37.A.emphasize               B.realize             C.decide             D.admit

38.A.envies                           B.blames             C.helps               D.hurts

39.A.noblest                   B.cruellest           C.most selfless     D.most diligent

40.A.phoned                   B.drove                     C.went                D.stayed

41.A.health                            B.business           C.experiment       D.treatment

42.A.better                    B.harder             C.easier                     D.healthier

43.       A.meanwhile              B.anyway            C.however          D.therefore

44.A.change                   B.incident           C.achievement     D.problem

45.A.belief                            B.carelessness      C.lies                  D.excuses

46.A.old                         B.new                 C.beautiful          D.cheap

47.A.one-third                B.half                 C.thing               D.closet

48.A.so                          B.but                  C.since               D.although

49.A.holes                      B.stains               C.hope                D.importance

50.A.woke                        B.warmed           C.cheered            D.tore

51.A.act                         B.shake               C.cry                  D.cough

52.A.still                        B.even                C.merely             D.seldom

53.A.somebody               B.anybody          C.everybody        D.nobody

54.A.controlling                     B.pushing           C.criticizing        D.fighting

55.A.beg                        B.allow               C.expect             D.cause