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Math Help - detriments problems someone please cheack to see if i am doing it right !

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    detriments problems someone please cheack to see if i am doing it right !

    |1 o |
    | 0 2 |

    = (2x1)-(0x0)=2

    |1 -2|
    |-1 3|
    = (1x3)-(-2x-1)=1

    | 2 -3 -2|
    | -1 3 4 |
    | -7 -2 8 |

    2| 3 4 | X 3 | -1 4 | x -2 | -1 3|
    |-2 8| | -7 8 | |-7 -2|

    = 2(8-24)+3(8+28)-2(2+21)

    = 94



    did i do these right ?
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by arsenal12345 View Post
    |1 o |
    | 0 2 |

    = (2x1)-(0x0)=2

    |1 -2|
    |-1 3|
    = (1x3)-(-2x-1)=1

    | 2 -3 -2|
    | -1 3 4 |
    | -7 -2 8 |

    2| 3 4 | X 3 | -1 4 | x -2 | -1 3|
    |-2 8| | -7 8 | |-7 -2|

    = 2(8-24)+3(8+28)-2(2+21)

    = 94



    did i do these right ?
    1 and 2 are correct. You need to remember that the signs alternate when evaluating a 3x3 determinant or higher. A diagram like this helps:

    \displaystyle \begin{align*} \left| \begin{matrix} + & - & + & - & \dots \\ - & + & - & + & \dots \\ + & - & + & - & \dots \\ - & + & - & + & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots  \end{matrix} \right| \end{align*}

    So to evaluate your 3x3 determinant, it would be

    \displaystyle \begin{align*} \left| \begin{matrix} \phantom{-}2 & -3 & -2 \\ -1 & \phantom{-}3 & \phantom{-}4 \\ -7 & -2 & \phantom{-}8   \end{matrix} \right| &= 2\left| \begin{matrix} \phantom{-}3 & 4 \\ -2 & 8 \end{matrix} \right| - (-3) \left| \begin{matrix} -1 & 4 \\ -7 & 8 \end{matrix} \right|  + (-2) \left| \begin{matrix} -1 & \phantom{-}3 \\ -7 & -2 \end{matrix} \right| \end{align*}
    Thanks from arsenal12345
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by Prove It View Post
    1 and 2 are correct. You need to remember that the signs alternate when evaluating a 3x3 determinant or higher. A diagram like this helps:

    \displaystyle \begin{align*} \left| \begin{matrix} + & - & + & - & \dots \\ - & + & - & + & \dots \\ + & - & + & - & \dots \\ - & + & - & + & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots  \end{matrix} \right| \end{align*}

    So to evaluate your 3x3 determinant, it would be

    \displaystyle \begin{align*} \left| \begin{matrix} \phantom{-}2 & -3 & -2 \\ -1 & \phantom{-}3 & \phantom{-}4 \\ -7 & -2 & \phantom{-}8   \end{matrix} \right| &= 2\left| \begin{matrix} \phantom{-}3 & 4 \\ -2 & 8 \end{matrix} \right| - (-3) \left| \begin{matrix} -1 & 4 \\ -7 & 8 \end{matrix} \right|  + (-2) \left| \begin{matrix} -1 & \phantom{-}3 \\ -7 & -2 \end{matrix} \right| \end{align*}
    man can you just work out the answear for me and explain it every step. i am lost !
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by arsenal12345 View Post
    man can you just work out the answear for me and explain it every step. i am lost !
    You know how to evaluate a 2x2 determinant, you showed this in questions 1 and 2. All you need to do here is evaluate these three 2x2 determinants and then perform the required multiplications, additions and subtractions...
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by Prove It View Post
    You know how to evaluate a 2x2 determinant, you showed this in questions 1 and 2. All you need to do here is evaluate these three 2x2 determinants and then perform the required multiplications, additions and subtractions...
    what is the requiered sub and multiplications ? thats the part i dont get
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by arsenal12345 View Post
    what is the requiered sub and multiplications ? thats the part i dont get
    Start by evaluating each of the 2x2 determinants...
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    Re: detriments problems someone please cheack to see if i am doing it right !

    Quote Originally Posted by Prove It View Post
    Start by evaluating each of the 2x2 determinants...
    ok after that what ? when does the signs alternate ?
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    Re: detriments problems someone please cheack to see if i am doing it right !

    man look can you just do a problem with solution please ? that will be more help full for me
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    Re: detriments problems someone please cheack to see if i am doing it right !

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    Re: detriments problems someone please cheack to see if i am doing it right !

    there are 2 ways to evaluate a 3x3 determinant:

    way #1) expansion by minors:

    \det(A) = a_{11}\det(A_{1,1}) - a_{12}\det(A_{1,2}) + a_{13}\det(A_{1,3})

     = a_{11}\det(A_{1,1}) - a_{21}\det(A_{2,1}) + a_{31}\det(A_{3,1})

    where A_{i,j} is the 2x2 matrix obtained by removing row i and column j.

    way #2) the rule of sarrus:

    \det(A) = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{12}a_{21}a_{33} - a_{11}a_{23}a_{32}

    (often displayed as three diagonal products each way that "wrap around")

    let us calculate:

    \det(A) = \begin{vmatrix}1&3&-2\\5&1&4\\0&2&1 \end{vmatrix}

    using both methods.

    the first way, we have a choice: expansion by a row, or by a column (we could use ANY row or column, but we'll use the first one to keep it simple). first we'll expand by the first row:

    to do this we need to find the 2x2 determinants, first.

    \det(A_{1,1}) = \begin{vmatrix} 1&4\\2&1 \end{vmatrix} = 1 - 8 = -7

    \det(A_{1,2}) = \begin{vmatrix} 5&4\\0&1 \end{vmatrix} = 5 - 0 = 5

    \det(A_{1,3}) = \begin{vmatrix} 5&1\\0&2 \end{vmatrix} = 10 - 0 = 10

    thus \det(A) = (1)(-7) - (3)(5) + (-2)(10) = -7 - 15 - 20 = -42.

    now we'll expand by the first column. we need to calculate 2 more 2x2 determinants (we'll re-use the one obtained by eliminating row 1 and column one):

    \det(A_{2,1}) = \begin{vmatrix} 3&-2\\2&1 \end{vmatrix} = 3 - (-4) = 7

    \det(A_{3,1}) = \begin{vmatrix} 3&-2\\1&4 \end{vmatrix} = 12 -(-2) = 14

    thus \det(A) = (1)(-7) - 5(7) + 0(14) = -7 - 35 + 0 = -42.

    finally, we'll use the rule of sarrus:

    \det(A)

     = (1)(1)(1) + (3)(4)(0) + (-2)(5)(2) - (-2)(1)(0) - (3)(5)(1) - (1)(4)(2)

     = 1 + 0 - 20 - 0 - 15 - 8 = 1 - 43 = -42.

    personally, i like the rule of sarrus, but expansion by minors generalizes better to larger matrices.
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