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Thread: Find the determminant?

  1. #1
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    Find the determminant?

    find the determinant
    0 1 -2
    -2 2 0
    -3 4 1


    find the inverse
    4 -5
    1 -1


    find the deterinant
    -4 3
    5 0
    Last edited by mr fantastic; Dec 13th 2010 at 05:17 PM. Reason: Title.
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  2. #2
    MHF Contributor
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    I expanded across the first row.

    $\displaystyle \displaystyle
    \begin{vmatrix}
    0 & 1 & -2\\
    -2 & 2 & 0\\
    -3 & 4 & 1
    \end{vmatrix}\Rightarrow 0*\begin{vmatrix}
    2 & 0 \\
    4 & 1
    \end{vmatrix}-1*\begin{vmatrix}
    -2 & 0 \\
    -3 & 1
    \end{vmatrix}-2*\begin{vmatrix}
    -2 & 2 \\
    -3 & 4
    \end{vmatrix}=0-1*(-2-0)-2*(-8-(-6))=0+2+4=6$

    You should be able to do this one by using 1 and 3 as an outline.

    $\displaystyle \displaystyle A^{-1}=\frac{1}{det(A)}*A$

    $\displaystyle \displaystyle\begin{vmatrix}
    -4 & 3 \\
    5 & 0
    \end{vmatrix}=-4*0-5*3=0-15=-15$

    Definition:
    The determinant of an n x n matrix A is defined as

    $\displaystyle \displaystyle
    det(A) =
    \begin{cases}
    a_{11}, & \mbox{if } \ n=1 \\
    a_{11}A_{11}+a_{12}A_{12}+\dots +a_{1n}A_{1n}, & \mbox{if } \ n>1
    \end{cases}
    $

    where

    $\displaystyle \displaystyle A_{1j}=(-1)^{1+j}det(M_{1j}) \ j=1,2,\cdots , n$

    are the cofactors associated with the first row expansion.
    Last edited by dwsmith; Dec 13th 2010 at 02:50 PM.
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