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Math Help - help needed!!! proof of invertible matrices

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    help needed!!! proof of invertible matrices

    The question has been attached as an image.

    any help or light on this question would be hugely appreciated!!

    thank you
    Attached Thumbnails Attached Thumbnails help needed!!! proof of invertible matrices-question-1.jpg  
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  2. #2
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    Re: help needed!!! proof of invertible matrices

    Quote Originally Posted by kaya2345 View Post
    The question has been attached as an image.
    any help or light on this question would be hugely appreciated!!
    thank you

    If \delta _{j,k}  = \left\{ {\begin{array}{rl}   {1,} & {j = k}  \\   {0,} & {j \ne k}  \\ \end{array} } \right. and I_n=\left[\delta _{j,k}\right].

    Is I_n invertible?

    Can we take away any of those ones?
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    Re: help needed!!! proof of invertible matrices

    Quote Originally Posted by Plato View Post
    If \delta _{j,k}  = \left\{ {\begin{array}{rl}   {1,} & {j = k}  \\   {0,} & {j \ne k}  \\ \end{array} } \right. and I_n=\left[\delta _{j,k}\right].

    Is I_n invertible?

    Can we take away any of those ones?
    thank you. and yes In is invertible
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    Re: help needed!!! proof of invertible matrices

    That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number. We do know that if a matrix has two columns the same, it is not invertible. I think we can avoid that by having a 0, at a different row, in each column. That implies that the maximum number of 1s in such an invertible n by n matrix is n^2- n.
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    Re: help needed!!! proof of invertible matrices

    Quote Originally Posted by HallsofIvy View Post
    That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number. We do know that if a matrix has two columns the same, it is not invertible. I think we can avoid that by having a 0, at a different row, in each column. That implies that the maximum number of 1s in such an invertible n by n matrix is n^2- n.
    Thank you, for that. Could I prove this using proof by induction? or contradiction?
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    Re: help needed!!! proof of invertible matrices

    Quote Originally Posted by HallsofIvy View Post
    That shows, I think, that the minimum number of 1s in such an invertible n by n matrix is n. But the question was about the maximum number.
    That mistake was once caused by my laptop not doing well with those attachments.
    I wish we could outlaw the posting of questions in images.
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