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Linear algebra problem

  1. Sep 17, 2008 #1
    Can someone help me proove this one please?

    If M is an nxn matrix

    _____|x+L x x . . . x |
    _____| x x+L x . . . x |
    |M|= | x x x+L. . . x | = [L^(n-1)]*(nx+L)
    _____| : : : : : : : : |
    _____| x x x . . . x+L|
     
  2. jcsd
  3. Sep 17, 2008 #2

    Defennder

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    Homework Helper

    What are x and L supposed to be here?
     
  4. Sep 17, 2008 #3
    I guess real numbers.
     
  5. Sep 17, 2008 #4
    Solved!!!!

    Abstract the n-1 column from the n column, then the n-2 from the n-1, then the n-3 from the n-2 etc.You get:

    |x+L -L 0 0 . . .0 0 0 |
    | x L -L 0 . . .0 0 0|
    | x 0 L -L . . .0 0 0|
    | x 0 0 L . . .0 0 0 |
    | . . . . . . . . . . . |
    | . . . . . . . . . . . |
    | x . . . . . .L -L 0|
    | x . . . . . .0 L -L|
    | x . . . . . .0 0 L|

    Then add The n row to the n-1, then the n-1 to n-2 etc and you get:

    | nx+ L 0 0 0 . . . 0 0 0|
    | (n-1)x L 0 0 . . .0 0 0|
    | (n-2)x 0 L 0 . . .0 0 0|
    | (n-3)x 0 0 L . . .0 0 0|
    | . . . . . . . . . . . . . . .| = (nx+L)*L^(n-1)
    | . . . . . . . . . . . . . . .|
    | 3x . . . . . . . . . L 0 0|
    | 2x . . . . . . . . . 0 L 0|
    | x . . . . . . . . . . 0 0 L|
     
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