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Thread: Showing that a matrix is not invertible

  1. #1
    Senior Member
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    Showing that a matrix is not invertible

    The question:
    Show that the following matrix is not invertible

    $\displaystyle \[
    M =
    \left[ {\begin{array}{ccc}
    \frac{-5}{3} &\frac{1}{3} & \frac{5}{3} \\
    \frac{13}{3} &\frac{10}{3} & \frac{5}{3} \\
    \frac{-7}{3} &\frac{-7}{3} & \frac{-5}{3} \\
    \end{array} } \right]
    \]$

    My attempt:
    I know that when the determinate is 0, a matrix is not invertible. So I did the following:

    $\displaystyle \[
    \left| {\begin{array}{ccc}
    \frac{-5}{3} &\frac{1}{3} & \frac{5}{3} \\
    \frac{13}{3} &\frac{10}{3} & \frac{5}{3} \\
    \frac{-7}{3} &\frac{-7}{3} & \frac{-5}{3} \\
    \end{array} } \right|
    \]$

    =
    $\displaystyle
    +
    \[
    \left| {\begin{array}{cc}
    \frac{13}{3} &\frac{10}{3} \\
    \frac{-7}{3} &\frac{-7}{3} \\
    \end{array} } \right|
    \]
    \[
    \left| {\begin{array}{cc}
    \frac{10}{3} &\frac{5}{3} \\
    \frac{-7}{3} &\frac{-5}{3} \\
    \end{array} } \right|
    \]
    -
    \[
    \left| {\begin{array}{cc}
    \frac{13}{3} &\frac{5}{3} \\
    \frac{-7}{3} &\frac{-5}{3} \\
    \end{array} } \right|
    \]
    $

    Which becomes:

    $\displaystyle (\frac{-50}{9} - (\frac{-35}{9}) - (\frac{-65}{9} - (\frac{-35}{9})) + (\frac{-91}{9} - (\frac{-70}{9}))$

    =

    $\displaystyle \frac{-15}{9} - \frac{-30}{9} + \frac{-21}{9} = \frac{-2}{3}$

    But this isn't 0 like required. :/

    What have I done wrong? Thanks.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Yuma, AZ, USA
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    for simplicity factor a $\displaystyle \frac{1}{3}$ out of the matrix and then add row 3 to both row 1 and row 2 to get

    $\displaystyle \frac{1}{3}\begin{vmatrix} -12 & -6 & 0 \\ 6 & 3 & 0 \\ -7 & -7 & -5\end{vmatrix}$

    Now expand down the third column to get

    $\displaystyle -5(-12(3)-6(-6))=-5(0)=0$
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  3. #3
    Senior Member
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    Thank you, I'll try that technique. I wonder why my way didn't work.
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
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    Yuma, AZ, USA
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    The formatting on your post is a bit garbled but I don't see any coeffeints on your minors. That may be where the problem is.
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  5. #5
    Senior Member
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    Oh, I'm such an idiot. >_<

    Thanks.
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