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Math Help - linear Algebra: Properties of Determinants

  1. #1
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    linear Algebra: Properties of Determinants

    Please could anyone explain this in details because my teacher did not explain this problem in detail in class. She just gave us the answer. I would like to know how to solve this in case she puts a similar problem on a test. Thanks.

    A equals
    0 1 2 3
    1 1 1 1
    -2 -2 3 3
    1 2 -2 -3

    Det (A) is equal to 10.

    Use the valuse of det (A) to evaluate

    0 1 2 3
    -2 -2 3 3
    1 2 -2 -3
    1 1 1 1

    +

    0 1 2 3
    1 1 1 1
    -1 -1 4 4
    2 3 -1 -2
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  2. #2
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    The second matrix answer is found by Det(2nd matrix)/10, and the third matrix answer is found by Det(3rd matrix)/10. So if you just plug those numbers in on your graphing calculator in the matrix section, and find the determinant of the matrix, you should be able to enter the equation shown above. Once you get the answers to both of the matrices, just add them together for your final answer. Sorry if this seems cluttered, its the best I could do. Hope this helped.
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  3. #3
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    If I understand the question correctly, you are given that det(A) = 10, where A = \begin{bmatrix} 0   &1&   2   &3\\ 1&1    &1&    1\\ -2 &-2&    3    &3\\1&    2  &-2   &-3 \end{bmatrix}. You have to evaluate the determinants \begin{vmatrix} 0 &1& 2 &3\\ -2 &-2& 3 &3\\1& 2 &-2 &-3 \\ 1&1 &1& 1\end{vmatrix} and \begin{vmatrix} 0 &1& 2 &3 \\ 1&1 &1& 1\\ -1 &-1& 4 &4\\2& 3 &-1 &-2\end{vmatrix} (and then add them together).

    You'll notice that the first of these has the same rows as A, but in a different order. You're supposed to know that if you interchange two rows of a matrix then this changes the sign of its determinant. So you have to work out how many row interchanges are needed in order to move the second row of A down to the bottom row. That will tell you the value of the first of those determinants.

    For the other determinant, you'll notice that the top two rows are the same as in A, but the bottom two rows have been changed a bit. You're supposed to know that if you add a multiple of one row of a matrix to another row then the determinant is unchanged. Is that enough of a hint?
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