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Math Help - Given that Matrix "A"=6 find ...this matrix

  1. #1
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    Given that Matrix "A"=6 find ...this matrix

    Given that the 3 x3 matrix:

    |a b c|
    |d e f|=6
    |g h i|

    find the 3x3 matrix:

    |-3b -3a -3c/2|
    | e d f/2|
    | h-4e g-4d (i-4f)/2|

    Any clue how to begin this question?
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Jishent View Post
    Given that the 3 x3 matrix:

    |a b c|
    |d e f|=6
    |g h i|

    find the 3x3 matrix:

    |-3b -3a -3c/2|
    | e d f/2|
    | h-4e g-4d (i-4f)/2|

    Any clue how to begin this question?
    Use elementary matrices for example first define

    A=\begin{bmatrix}a & b & c \\ d & e &f \\ g & h& i \end{bmatrix}

    Then use

    E_1=\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 &0 \\ 0 & 0& 1 \end{bmatrix}

    Multiplying by this matrix has the effect of swapping column 1 and column 2

    AE_1=\begin{bmatrix}b & a & c \\ e & d &f \\ h & g& i \end{bmatrix}

    Now just come up with the other elementary matrices to do the rest of your row or column operations and use the fact that

    \det(AB)=\det(A)\cdot \det(B)
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  3. #3
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    TheEmptySet is right; basically, you are applying elementary row (or column) operations to the first matrix to get the second one. Each row operation does something specific to the determinant.

    I just wanted to ask something that wasn't clear to me, from the way that you asked the question...

    You ARE aware that they are asking for determinants, correct? Just asking because I didn't see you use the word anywhere, but you did say stuff like "Matrix A = 6", which doesn't really make any sense.
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  4. #4
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    I am still confused what to do after I times AE matrices. Can you elaborate more then? Thanks!
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  5. #5
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    Do another elementary row operation: row swapping, adding or subtracting rows, or multiplying by a scalar. Keep doing that until you obtain the 2nd matrix.

    Also note, adding/subtracting rows has no affect on the determinant; multiplication scales the determinant by the factor of x; and swapping multiplies the det by -1.
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  6. #6
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    Thanks

    Thanks everyone! BTW I have one more question.

    Is the determinant of matrix A equal to the determinant of matrix A transpose. if and only if A is symmetric. (Both square matrices.)
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  7. #7
    MHF Contributor harish21's Avatar
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  8. #8
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    Quote Originally Posted by Jishent View Post
    Thanks everyone! BTW I have one more question.

    Is the determinant of matrix A equal to the determinant of matrix A transpose. if and only if A is symmetric. (Both square matrices.)
    No, they're always equal, symmetric or not.
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