? asked in 科學數學 · 1 decade ago

這是一題有關calculus的問題 拜託幫幫我~

Suppose A is a 3 x 3 matrix such that A^4- 2A^3 + 5A^2 - 2I = 0. Show that A is invertible.

(Hint: think carefully about the defnition of invertibility of a matrix on page 264).

definition: if A is a square matrix and there exists a matrix C such that CA=I, then C is called

可以幫我求出來嗎? 拜託拜託>"<

Update:

then C is called an inverse of A, and A is said to be INVERTIBLE

2 Answers

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  • linch
    Lv 7
    1 decade ago
    Favorite Answer

    A^4- 2A^3 + 5A^2 - 2I = 0 ==> A^4- 2A^3 + 5A^2 = 2I

    A ( A^3 - 2A^2 + 5A ) = 2I ==> A [ (1/2) A^3 - A^2 + (5/2) A ) = I

    所以 (1/2) A^3 - A^2 + (5/2) A 是 A 的 Inverse

    因此 A 是 invertible

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  • Anonymous
    6 years ago

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