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Math Help - Determinant of a matrix using cofactor expansion - confused

  1. #1
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    Determinant of a matrix using cofactor expansion - confused

    Hey! I am studying for my linear algebra exam and am having some trouble with determining the determinant of a square matrix using cofactor expansion.

    Here is an example of what I am having trouble with:
    Question
    \displaystyle \begin{bmatrix}<br />
0 & 3 & 1\\ <br />
1 & 1 & 2\\ <br />
3 & 2 & 4\\<br />
\end{bmatrix}<br />

    Attempted solution
    I know that the determinant of this matrix is 5, I just can't seem to get that answer in what I am doing. I will use the first row in this attempt:
     0 * \displaystype \begin{bmatrix}<br />
1 & 2\\<br />
2 & 4\\ \end{bmatrix}<br />
     + 3 * \displaystype \begin{bmatrix}<br />
1 & 2\\<br />
3 & 4\\ \end{bmatrix}<br />
     + 1 * \displaystype \begin{bmatrix}<br />
1 & 1\\<br />
3 & 2\\ \end{bmatrix}<br />
    then replacing those 2x2 matrices with their determinants gives:
     (0 * 0) + (3 * -2) + (1 * -1)
     = -7
    Which is NOT 5. I tried this with other columns and rows and am getting different answers for those. Am I doing this wrong or does it not work on some matrices???

    Hopefully someone can help! Thanks for looking
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by Kakariki View Post
    Hey! I am studying for my linear algebra exam and am having some trouble with determining the determinant of a square matrix using cofactor expansion.

    Here is an example of what I am having trouble with:
    Question
    \displaystyle \begin{bmatrix}<br />
0 & 3 & 1\\ <br />
1 & 1 & 2\\ <br />
3 & 2 & 4\\<br />
\end{bmatrix}<br />

    Attempted solution
    I know that the determinant of this matrix is 5, I just can't seem to get that answer in what I am doing. I will use the first row in this attempt:
     0 * \displaystype \begin{bmatrix}<br />
1 & 2\\<br />
2 & 4\\ \end{bmatrix}<br />
    -   3 * \displaystype \begin{bmatrix}<br />
1 & 2\\<br />
3 & 4\\ \end{bmatrix}<br />
     + 1 * \displaystype \begin{bmatrix}<br />
1 & 1\\<br />
3 & 2\\ \end{bmatrix}<br />
    then replacing those 2x2 matrices with their determinants gives:
     (0 * 0) \bold{-}  (3 * -2) + (1 * -1)
     = 5
    Which is NOT 5. I tried this with other columns and rows and am getting different answers for those. Am I doing this wrong or does it not work on some matrices???

    Hopefully someone can help! Thanks for looking
    The sign of the co-factors is changing from column to column (or row to row).

    See my corrections.
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  3. #3
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    I looked back over my notes and I seemed to have somehow missed the part explaining the pattern of sign changing... With that I understand how to go about doing this!!! Thank you so much!
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