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Math Help - n rows determinant value 6

  1. #1
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    n rows determinant value 6

    \begin{vmatrix}<br />
1 &.  &.  &.  &  &1 \\ <br />
 0&1  &  &  .& . & 1\\ <br />
 1&  &  2&  &  &1 \\ <br />
 .&  &  & 3&  & 1\\ <br />
 .&  &  &  & . & 1\\ <br />
 1&  &  &  &  & n-1<br />
\end{vmatrix}=-(n-2)!

    is it true
    Last edited by transgalactic; November 16th 2010 at 11:13 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by transgalactic View Post
    \begin{vmatrix}<br />
1 &.  &.  &.  &  &1 \\ <br />
 0&1  &  &  .& . & 1\\ <br />
 1&  &  2&  &  &1 \\ <br />
 .&  &  & 3&  & 1\\ <br />
 .&  &  &  & . & 1\\ <br />
 1&  &  &  &  & n-1<br />
\end{vmatrix}=-(n-2)!

    is it true
    It's not clear what the values that are "dotted" are supposed to be. One? Zero? Regardless, the easiest (i.e. least clever) is to proceed by induction and use partitioned matrix multiplication.
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  3. #3
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    dots means that 1 appears n-1 times to the length and to the height
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  4. #4
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    from row reduction i got
    \begin{vmatrix}<br />
1 &1  &... &  &  &1 \\ <br />
 -1&0  & 0 &...& 0 & 0\\ <br />
 0& 0 &  1& 0.. &  &0 \\ <br />
 .& 0 &0  & 2& .. & 0\\ <br />
 .&  0&0  &0  & . & 0\\ <br />
 0& 0 &..  &  &  & n-1<br />
\end{vmatrix}

    how to find this determinant
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  5. #5
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    Expand on the left column.
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