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Math Help - invertible matrix

  1. #1
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    invertible matrix

    I have some difficulty proving the bolded part of the exercise:

    If A is an nxn matrix establish the identity
    In-Ak+1=(In-A)(In+A+A2+...+Ak).
    Deduce that if some power of A is the zero matrix then In-A is invertible.
    Suppose now that
    A=2 2 -1 -1
    -1 0 0 0
    -1 -1 1 0
    0 1 -1 1
    Compute the powers (In-A)i for i=1,2,3,4 and, by considering
    A=I4-(I4-A), prove that A is invertible and determine A-1.

    I would be grateful for any help you are able to provide...
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  2. #2
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    Re: invertible matrix

    The problem seems to suggest that (Iₙ - A)⁴ = 0, but it turns out that no power of (Iₙ - A) is 0. So I am not sure how to use the first part of the problem. Is this your difficulty?
    Thanks from jojo7777777
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  3. #3
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    Re: invertible matrix

    The equation (Iₙ - A)⁴ = 0 is correct....(I computed it here Online Matrix Calculator) so I still have the same problem...
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  4. #4
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    Re: invertible matrix

    Hmm, I am not sure what the problem is with WolframAlpha, but you are right.

    Have you done the first part where you were told to deduce the following fact:

    If some power of B is the zero matrix then Iₙ - B is invertible (*)

    (I replaced A with B)? If yes, then you put B := I - A for this particular A. You verified that B⁴ = 0; therefore, by (*) we have I - B = I - (I - A) = A is invertible. Moreover, by the first part of the problem,

    (I - B)(I + B + B + B) = I - B⁴ = I,

    so the inverse of I - B = A is I + B + B + B.
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  5. #5
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    Re: invertible matrix

    Thank you...you are genius!
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