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Math Help - 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

    \[<br />
M =<br />
\left[ {\begin{array}{ccc}<br />
 \frac{-5}{3} &\frac{1}{3}  & \frac{5}{3}  \\<br />
 \frac{13}{3} &\frac{10}{3}  & \frac{5}{3}  \\<br />
 \frac{-7}{3} &\frac{-7}{3}  & \frac{-5}{3}  \\<br />
 \end{array} } \right]<br />
\]

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

    \[<br />
\left| {\begin{array}{ccc}<br />
 \frac{-5}{3} &\frac{1}{3}  & \frac{5}{3}  \\<br />
 \frac{13}{3} &\frac{10}{3}  & \frac{5}{3}  \\<br />
 \frac{-7}{3} &\frac{-7}{3}  & \frac{-5}{3}  \\<br />
 \end{array} } \right|<br />
\]

    =
    <br />
+<br />
\[<br />
\left| {\begin{array}{cc}<br />
 \frac{13}{3} &\frac{10}{3}  \\<br />
 \frac{-7}{3} &\frac{-7}{3}  \\<br />
 \end{array} } \right|<br />
\]<br />
\[<br />
\left| {\begin{array}{cc}<br />
 \frac{10}{3} &\frac{5}{3}  \\<br />
 \frac{-7}{3} &\frac{-5}{3}  \\<br />
 \end{array} } \right|<br />
\]<br />
 -<br />
\[<br />
\left| {\begin{array}{cc}<br />
 \frac{13}{3} &\frac{5}{3}  \\<br />
 \frac{-7}{3} &\frac{-5}{3}  \\<br />
 \end{array} } \right|<br />
\] <br />

    Which becomes:

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

    =

    \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
    Joined
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    Yuma, AZ, USA
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    for simplicity factor a \frac{1}{3} out of the matrix and then add row 3 to both row 1 and row 2 to get

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

    Now expand down the third column to get

    -5(-12(3)-6(-6))=-5(0)=0
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  3. #3
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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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    Thanks
    78
    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
    Joined
    Apr 2010
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    Oh, I'm such an idiot. >_<

    Thanks.
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