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Math Help - Prove that matrix A is singular

  1. #1
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    Prove that matrix A is singular

    I am not sure where to begin with a question like this:

    Question: Let A be an n * n matrix. Prove that A is singular if and only if lambda = 0 is an eigenvalue of A.

    Any guidance will be appreciated.
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  2. #2
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    Quote Originally Posted by sparky View Post
    I am not sure where to begin with a question like this:

    Question: Let A be an n * n matrix. Prove that A is singular if and only if lambda = 0 is an eigenvalue of A.

    Any guidance will be appreciated.
    Start with

    Ax=\lambda x

    If you need more guidance,

    Ax-\lambda x=(A-I\lambda)x=0

    You will have to go the other direction after this too.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    An alternative:

    A\;\textrm{singular}\Leftrightarrow\det A=0 \Leftrightarrow \det (A-0I)=0\Leftrightarrow 0\;\textrm{is\;eigenvalue\;of\;}A


    Fernando Revilla
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