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Math Help - Determinant of a matrix with all-zero diagonal

  1. #1
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    Determinant of a matrix with all-zero diagonal

    "Given a nxn matrix with all of the entries along the main diagonal equal to zero, and every off-diagonal entry equal to one, compute its determinant"

    This is a question from a past exam paper I was working on. I understand that the determinant is equal to (n-1).(-1)^{n+1}, but I only got this from computing the inverse of 2x2, 3x3, 4x4, and 5x5 matrices of similar form, until I saw th pattern and then generalised, but I don't think this would be an acceptable method in an exam. I'm trying to find a wat to formally prove this, but all I could think of was proof by induction, and I couldn't work that out. Am I going at it the right way by induction, or is there another method?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Using the transformations

    (i)\;R_2\to R_2-R_1\;,\;\ldots\;,\;R_n\to R_n-R_1

    (ii)\; C_1\to C_1+C_2+\ldots+C_n

    you'll obtain a triangular determinant.
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  3. #3
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    Looks like a good job for induction. 2 and 3 are easily computed. Expand 4x4 by minors and you'll have a collection of 3x3s. Expand 5x5 by minors and get a collection of 4x4s. Let's see what you get.
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