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矩陣,A=2I+B和AB=O,若(A+I)^8=kA+I,求k=?

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矩陣,A=2I+B和AB=O,若(A+I)^8=kA+I,求k=?

已知 A=2I+B  和 AB=O, 若(A+I)^8=kA+I   求k     感恩

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\(A=2I+B \Rightarrow B = A-2I\)

\(\Rightarrow AB = A^2-2A=0\)

\(\Rightarrow A^2 = 2A \Rightarrow A^n = 2^{n-1} A,\,\forall n\in\mathbb{N}\)

\(\displaystyle\left(A+I\right)^8 = C^8_0 A^8 + C^8_1 A^7+\cdots+C^8_7A + C^8_8 I\)

  \(\displaystyle= 2^7 C^8_0A + 2^6 C^8_1 A+\cdots + C^8_7 A + I\)

  \(\displaystyle= \frac{1}{2}\left(2^8 C^8_0 + 2^7 C^8_1+\cdots 2C^8_7 +C^8_8 - 1\right) A +I\)

  \(\displaystyle= \frac{1}{2}\left(3^8 - 1\right) A +I\)

  \(\displaystyle= 3280 A+I\)

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引用:
原帖由 weiye 於 2016-7-3 08:50 PM 發表
\(A=2I+B \Rightarrow B = A-2I\)

\(\Rightarrow AB = A^2-2A=0\)

\(\Rightarrow A^2 = 2A \Rightarrow A^n = 2^{n-1} A,\,\forall n\in\mathbb{N}\)

...
感謝老師.....

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