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Thread: Linear Algebra Identity Understanding

  1. #1
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    Linear Algebra Identity Understanding

    I have a solution here that I don't quite understand.


    Q: Show that if a square matrix A satisfies the equation A^2+2A+I=0 , then A must be invertible.

    A:
    A^2+2A=-I
    -A^2-2A=I
    I got this far on my own, but the next step confuses me...
    A(-A-2I)=I

    I get that if A^-^1=(-A-2I) then A*A^-^1=I, however what I'm confused about is when they draw out the A from -2A.

    Could someone help me understand why A/A=I?

    Thank you!
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  2. #2
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    Re: Linear Algebra Identity Understanding

    Quote Originally Posted by PodoTheGreat View Post
    Q: Show that if a square matrix A satisfies the equation A^2+2A+I=0 , then A must be invertible.
    A:
    A^2+2A=-I
    -A^2-2A=I
    I got this far on my own, but the next step confuses me...
    A(-A-2I)=I
    I get that if A^-^1=(-A-2I) then A*A^-^1=I, however what I'm confused about is when they draw out the A from -2A.
    Do you agree that that $A(-A-I)=-A^2-A~?$
    If so, then can you explain you difficulty?
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  3. #3
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    Re: Linear Algebra Identity Understanding

    Matrices obey the "distributive law": A(B+ C)= AB+ AC. That is what you are using when you factor.
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