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Math Help - Matrix- Determinan and Inverse

  1. #1
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    Matrix- Determinan and Inverse

    Hello MHF, please help me with this excercise.
    1-Assuming the identity matrix "I3x3" was obtained from a matrix "A3x3" by means of the following operations on the rows of A:
    L1 <-> L3;
    L3 -> L3 - 2*L2;
    L3 ->1/2*L3;
    a) Det(A)= ?
    b) A^(-1)= ?
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  2. #2
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    Re: Matrix- Determinan and Inverse

    Hi,
    Remember elementary row transformations can be accomplished by left multiplication of the corresponding elementary row matrix. The elementary row matrix corresponding to a row transformation is the row transformation applied to the identity matrix. Use these ideas for a solution:

    Matrix- Determinan and Inverse-mhflinearalgebra2.png
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  3. #3
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    Re: Matrix- Determinan and Inverse

    Let A be an n x n matrix. Remember these effects of elementary row operations on the determinant.

    If B is the matrix obtained from A by interchanging two rows, then det(B) = -det(A)
    If B is the matrix obtained from A by adding a mutliple of one row to another, then det(B) = det(A)
    If B is the matrix obtained from A by multiplying a row by a scalar c, then det(B) = c det(A)

    Let B1 be the matrix obtained from A by L1 <--> L3. Then det(B1) = -det(A).
    Let B2 be the matrix obtained from B1 by L3 --> L3 - 2 L2. Then det(B2) = det(B1)
    Let B3 be the matrix obtained from B2 by L3 --> 1/2 L3. The det(B3) = (1/2) det(B2)

    Since B3 is the final elementary row operation, then we have the identity matrix. Thus, B3 = In. Then det(B3) = 1

    det(B3) = (1/2) det(B2)
    det(B3) = (1/2) det(B1)
    det(B3) = (1/2) - det(A)
    1 = -1/2 det(A)
    det(A) = -2

    Use augmented matrix.

    Let A =
    [a b c | 1 0 0]
    [d e f | 0 1 0]
    [g h i | 0 0 1]

    Then B1 =
    [g h i | 0 0 1]
    [d e f | 0 1 0]
    [a b c | 1 0 0]

    Then B2 =
    [g h i | 0 0 1]
    [d e f | 0 1 0]
    [a-2d b-2e c-2f | 1 -2 0

    Then B3 =
    [g h i | 0 0 1]
    [d e f | 0 1 0]
    [.5a-d, .5b-2, .5c-2 | .5 -1 0]

    Since B3 = I3, we see that A^-1 =

    [0 0 1]
    [0 1 0]
    [.5 -1 0]
    Last edited by mathguy25; April 20th 2013 at 09:05 PM.
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  4. #4
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    Re: Matrix- Determinan and Inverse

    I got it, thank you
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