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Bizarre matrix problem

  1. Dec 6, 2009 #1
    I found this odd question in the back of the chapter supplement. I don't need to do it for homework and it probably won't be on the test but the fact that I don't know how to do the problem frustrates me.

    Here it is:

    * * *

    Use the fact that 21,375, 38,798, 34,162, 40,223, and 79,154 are all divisible by 19 to show that

    | 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 |

    is divisible by 19 without directly evaluating the determinant.

    * * *

    It wasn't too hard to notice that the digits of the numbers given were also the digits you get from reading off the numbers in the rows... but this seems like a very superficial relationship!
     
  2. jcsd
  3. Dec 6, 2009 #2
    Do you remember the property that the determinant of a matrix remain constant if you add a multiple of any column to any other column?
     
  4. Dec 6, 2009 #3
    No, but now I do. What are you getting at?
     
  5. Dec 6, 2009 #4
    Try to add some other columns to the last column.
     
  6. Dec 6, 2009 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The point being that 2(10000)+ 1(1000)+ 3(100)+ 7(10)+ 5= 21375, 3(10000)+ 8(1000) +7(100)+ 9(10)+ 8= 38798, etc. That particular linear combination of the numbers on any one row is divisible by the same number.
     
  7. Dec 6, 2009 #6
    Well, if you want to put it *THAT* bluntly...
     
  8. Dec 6, 2009 #7
    Thank you.
     
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