# 線性代數考古題請高手幫忙

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• 7 years ago

Consider the characteristic polynomial

|A-pI|

= (2-p)^2*(3-p)+2+2- 2(2-p) - 2(2-p) - (3-p)

= -p^3+7p^2-11p+5

So -A^3+7A^2-11A+5I = 0

Also, consider another polynomial (p^5-8p^4+19a^3-2a^2+18a-6)

by long division(or other possible means), we obtain

(p^5-8p^4+19a^3-2a^2+18a-6)

= (-p^2+p-1)(-p^3+7p^2-11p+5) + 7p^2+3p-1

So

A^5-8A^4+19A^3-2A^2+18A-6I

= (-A^2+A-I)(-A^3+7A^2-11A+5I) + 7A^2+3A-I

= 7A^2+3A - I

(The rest is left to you since multiplication of matrices is quite easy, and I don't want to type out those matrices, most importantly xD)

2014-01-26 23:28:10 補充：

the first part is the application of Cayley-Hamilton theorem.

2014-01-26 23:39:18 補充：

sorry for my wrong calculations during long division, it should be

(p^5-8p^4+19p^3-2p^2+18p-6)

= (-p^2+p-1)(-p^3+7p^2-11p+5) + 21p^2+2p-1

So

A^5-8A^4+19A^3-2A^2+18A-6I

= (-A^2+A-I)(-A^3+7A^2-11A+5I) + 21A^2+2A-I

= 21A^2+2A - I

• 7 years ago

萬分感謝！

數學跟科學是相當精準的知識，這一點很吸引我。希望能夠跨入這些領域的大門，一窺自然的奧秘。

• 7 years ago

人家已經說了「想考數學研究所」

^__^

2014-01-27 12:14:51 補充：

我也祝你成功

加油！加油！加油！

╭∧---∧╮

│ .✪‿✪ │

╰/) ⋈ (\╯

• 7 years ago

好奇想問一下，你擅長領域是心理學，那大學的專業應該是心理學吧？

心理學有牽涉到線性代數？

2014-01-26 23:23:55 補充：

原來如此。

祝你成功

加油！加油！加油！