已知矩阵
与对角矩阵Λ相似，求a的值，并求可逆矩阵P，使P-1AP=Λ．
参考答案：[解] 由[*]
得到矩阵A的特征值 λ1=λ2=3，λ3=-1．
由矩阵A的特征值有重根，而A与对角矩阵相似，可知λ=3必有2个线性无关的特征向量，因而秩r(3E-A)=1．于是由
[*]
对λ1=λ2=3，解齐次线性方程组(3E-A)x=0，
[*]
得基础解系：α1=(1，1，0)T，α2=(1，0，1)T．
对λ3=-1，解齐次线性方程组(-E-A)x=0，
[*]
得基础解系：α3=(1，-3，0)T．
令
[*]
| 阅读理解。 |
| Fred liked playing tricks on others. One afternoon he went to a shop and bought some bread. On his way home he saw Johnson, an old man about sixty riding a donkey (驴). The boy walked to the old man and decided to play a trick on him. He thought for a moment and then he had an idea. "Hello, my friend," Fred said in a loud voice. "Please come and have a piece of bread." "Thank you very much." Johnson got off the donkey and said gladly." "But I have already had lunch." "I'm sorry. I'm talking to your donkey, not to you." Fred said with a smile. Johnson understood at once that the boy was playing a trick on him. He became angry, but he didn't show his anger. He just turned quickly to his donkey, held up the lash (鞭子) and hit it on the back. "Before we set off, I asked if you had friends in the town. You said again and again that you hadn't. Can you explain why the young man asked you to have the bread, then?" The more the old man said, the less he could control his anger. He finally shouted, "I'll beat you to death if you dare fool me again!" Hearing this, Fred's face turned red and he left away quickly. |
| 1. Fred ______. |
[ ] |
| A. was clever but not polite. B. liked playing with donkeys C. often help others D. wasn't clever |
| 2. Johnson ______ at first. |
[ ] |
| A. beat his donkey to death B. was angry to meet the boy C. wanted to eat the bread D. thought the boy was talking to him |
| 3. Which of the following is true? |
[ ] |
| A. The old man didn't beat the donkey. B. The old man taught Fred how to be polite. C. The old man beat the donkey to death. D. The man played a trick back on the boy. |