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Math Help - Prove that the det of a Matrix is divisible by 19 given...

  1. #1
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    Prove that the det of a Matrix is divisible by 19 given...

    So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

    The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

    \begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4&  0&2&2&3\\7&9&1&5&4\\\end{pmatrix}

    is also divisible by 19

    Any help would be appreciated
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  2. #2
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    Quote Originally Posted by AlmightyMidget View Post
    So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

    The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

    \begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4&  0&2&2&3\\7&9&1&5&4\\\end{pmatrix}

    is also divisible by 19

    Any help would be appreciated
    do this column operation and you'll be done (here C_j is the j-th column): 10000C_1 + 1000C_2 + 100C_3 + 10C_4+C_5 \to C_5.
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  3. #3
    Senior Member pankaj's Avatar
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    How did tou figure it out? Is there a general way to do it
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  4. #4
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    Quote Originally Posted by pankaj View Post

    How did tou figure it out? Is there a general way to do it
    i think if you do what i suggested in my previous post, you'll see how i figured it out.
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  5. #5
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    Thanks very much NonCommAlg! I did what you suggested during my break today, and I almost kicked myself, it seems so simple, thanks!
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  6. #6
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    Quote Originally Posted by AlmightyMidget View Post
    So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

    The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

    \begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4&  0&2&2&3\\7&9&1&5&4\\\end{pmatrix}

    is also divisible by 19

    Any help would be appreciated
    Wait the 5 numbers are ordered pairs: 21,375 38,798 34,162 40,223 and 79,154?
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