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Math Help - 2 Linear Algebra Question

  1. #1
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    2 Linear Algebra Question




    1.Let B an nxn matrix such that B^4=0

    Show that A=I-B is invertible and A^-1 = I+B+B^2+B^3?

    2 Let A an nxn Matrix
    a.If A^2=0 show that A is not Invertible
    b. If A^2= A and A is not the identity, show that A is not Invertible.

    plz help.. i couldnt seem to figure it out?
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  2. #2
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    Quote Originally Posted by firebio View Post



    1.Let B an nxn matrix such that B^4=0

    Show that A=I-B is invertible and A^-1 = I+B+B^2+B^3?
    The definition of A^{-1} is that AA^{-1}= I and A^{-1}A= I

    (I- B)(I+ B+ B^2+ B^3)= I(I+ B+ B^2+ B^3)- B(I+ B+ B^2+ B^3) = (I+ B+ B^2+ B^3)-(B+ B^2+ B^3+ B^4)= I- B^4= I since B^4= 0.

    (I+ B+ B^2+ B^3)(I- B)= (I+ B+ B^2+ B^3)I- (I+ B+ B^2+ B^3)B = (I+ B+ B^2+ B^3)- (B+ B^2+ B^3+ B^4)= I- B^4= I.

    2 Let A an nxn Matrix
    a.If A^2=0 show that A is not Invertible
    b. If A^2= A and A is not the identity, show that A is not Invertible.

    plz help.. i couldnt seem to figure it out?
    a) (proof by contradiction) Suppose A were invertible. Then, multiplying both sides by A^{-1}, A^{-1}A^2= A= 0 which is a contradiction because 0 is not invertible.

    Do exactly the same thing for b.
    Last edited by HallsofIvy; July 21st 2009 at 07:36 AM.
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