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Find the determinant using row operations

  1. Apr 21, 2009 #1
    1. The problem statement, all variables and given/known data
    find the determinant using row operations:
    1 -2 2
    0 5 -1
    2 -4 1



    2. Relevant equations



    3. The attempt at a solution
    i took row 3 and took 2 x row 1 away from it to get :
    1 -2 2
    0 5 -1
    0 0 -3
    1 x 5 x (-3) = -15...but i multiplied a row by 2 so i should get -30 for the det right?but the answer in my book is -15..what am i doing wrong?
     
  2. jcsd
  3. Apr 21, 2009 #2
    Re: determinant

    Calculating the determinant for A, a 3x3 matrix with elements:
    a b c
    d e f
    g h i

    Det(A) = a(ei - fh) - b(di - fg) + c(dh-eg), by starting with row 1.

    So -15 is the answer you should be getting.
     
  4. Apr 21, 2009 #3
  5. Apr 21, 2009 #4

    Mark44

    Staff: Mentor

    Re: determinant

    Adding a row or a multiple of a row to another row doesn't change the value of the determinant. If you had replaced a row by a multiple of itself, then the determinant's value would have changed.
     
  6. Apr 21, 2009 #5
    Re: determinant

    hmmm well i looked over my book, and it said if i multiply a row by k...i multiply the determinant by k...so when do we multiply the determinant by k?..cause what i did was that not multiplying a row by k(2)?

    and thanks for ur help guys
     
  7. Apr 21, 2009 #6

    Mark44

    Staff: Mentor

    Re: determinant

    You did not replace a row by some multiple of itself; you added a multiple of a row to another row. These are different operations. On the other hand, if you had replaced row 1 by -2 times itself, and then added the first row to the third row, then your determinant would have been +30. This is not what you did though, since the first row stayed the same from start to finish.

    In one of the linear algebra books I have, there is a theorem about determinants and row operations. The theorem has three parts.
    1. If you interchange two rows, the determinant of the new matrix will be -1 times the determinant of the old matrix.
    2. If you replace a row by k times itself, the determinant of the new matrix will be k times the determinant of the old matrix.
    3. If you add k times one row to another row, the determinant of the new matrix will be equal to the determinant of the old matrix.
     
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