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Math Help - Determinant

  1. #1
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    Determinant

    |A| = \begin{vmatrix} 1 & 2&3&\dots&n-1&n\\-1 & 0&3&\dots & n-1&n\\-1&-2&0&\dots &n-1 &n\\ \vdots&&&&&\\ -1& -2& -3&\dots & -(n-1) & 0 \end{vmatrix}

    Show that |A|=n!.
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  2. #2
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    Quote Originally Posted by james_bond View Post
    |A| = \begin{vmatrix} 1 & 2&3&\dots&n-1&n\\-1 & 0&3&\dots & n-1&n\\-1&-2&0&\dots &n-1 &n\\ \vdots&&&&&\\ -1& -2& -3&\dots & -(n-1) & 0 \end{vmatrix}

    Show that |A|=n!.
    First, for 1\leqslant j\leqslant n you can take a factor j out of the j'th column, giving you |A| = n!\begin{vmatrix} 1 & 1&1&\dots&1&1\\-1 & 0&1&\dots & 1&1\\-1&-1&0&\dots &1 &1\\ \vdots&&&&&\\ -1& -1& -1&\dots & -1 & 0 \end{vmatrix}. Now add the top row of that new matrix to each of the other rows and you'll get an upper-triangular matrix with 1s on the diagonal. So its determinant is 1.
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