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Math Help - Help finding determinants of matrices

  1. #1
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    Help finding determinants of matrices

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

    find the determinants of these matrices:
    a)
    |a c b |
    B = |d f e |
    |g i h |

    b)2A
    c) A^-1 (inverse of A)

    Is there a way to find a number value for the determinants of these matrices only knowing that |A| = 6 ?
    I know how to find a determinant, but this question is confusing to me since there are no values
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hint for a:
    recall the theorem which state that if if you replace one column with another the det. isn't changes.(wrong!!!)

    for b.

    det(2A)=2det(A)

    for c.

    det(A)=det(A^-1)
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  3. #3
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    |A|= a(ei-fh)-b(di-fg)+c(dh-eg) =aei-afh-bdi+bfg+cdh-ceg=  6

    |B| =a(fh-ei)-c(dh-eg)+b(di-fg)  =-aei+afh+bdi-bfg-cdh+ceg= -(aei-afh-bdi+bfg+cdh-ceg)=-|A| = -6
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  4. #4
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    Sean, please post in laTEX. LaTEX makes your math readable.
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  5. #5
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    thanks to both of you. just to clarify, zarathustra is right for b and c ?
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  6. #6
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    Hello, seanP!

    \text{Suppose: }\;A \:=\:\left|\begin{array}{ccc} a&b&c \\ d&e&f \\ g&h&1\end{array}\right| \;=\;6

    Find the determinants of these matrices:

     a)\;B \;=\;\left|\begin{array}{ccc}a&c&b \\ d&f&e \\ g&i&h \end{array}\right|

    If two adjacent rows (or two adjacent columns) are interchanged,
    . . the sign of the determinant is changed.

    Therefore: . B \:=\:-6




    b)\;2A

    Too easy!

    2A \;=\;2(6)  \;=\;12




    c)\;A^{-1}

    \text{Since }A\cdot A^{-1} \:=\:I\,\text{ and }\,|I| \:=\:1

    . \text{we have: }\;6\cdot A^{-1} \:=\:1 \quad\Rightarrow\quad A^{-1} \:=\:\dfrac{1}{6}

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  7. #7
    Master Of Puppets
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    Hi Soroban, the OP has stated (not so clearly!) that

    |A| \:=\:\left|\begin{array}{ccc} a&b&c \\ d&e&f \\ g&h&1\end{array}\right| \;=\;6

    so

    A \:=\:\left[\begin{array}{ccc} a&b&c \\ d&e&f \\ g&h&1\end{array}\right] \implies 2A \:=\:2\left[\begin{array}{ccc} a&b&c \\ d&e&f \\ g&h&1\end{array}\right]

    Your solution is for 2|A|\neq 2A
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