2.9.1 Examples for the calculation rules for determinants

  1. A change of two columns switches the sign of \(\det(\tilde A)\)
    example:\(\;\;\left|\begin{array}{ccc}2&3&0\\0&0&1\\4&3&4\end{array}\right|=-\left|\begin{array}{ccc}2&0&3\\0&1&0\\4&4&3\end{array}\right|\)

  2. A common (scalar) factor of a column can be taken in front of \(\det(\tilde A)\)
    example:\(\;\;\left|\begin{array}{ccc}2&3&0\\0&0&1\\4&3&4\end{array}\right|=3\left|\begin{array}{ccc}2&1&0\\0&0&1\\4&1&4\end{array}\right|=6\left|\begin{array}{ccc}1&1&0\\0&0&1\\2&3&4\end{array}\right|\)

  3. Two determinants which are equal up to one column can be added according to
    example:\(\;\;\left|\begin{array}{ccc}2&3&0\\0&0&1\\4&3&4\end{array}\right|+\left|\begin{array}{ccc}2&1&0\\0&2&1\\4&3&4\end{array}\right|=\left|\begin{array}{ccc}2&3+1&0\\0&0+2&1\\4&3+3&4\end{array}\right|\)

  4. If a multiple of a column is added to another column the \(\det(\tilde A)\) is not changed,
    example:\(\;\;\left|\begin{array}{ccc}2&3&0\\0&0&1\\4&3&4\end{array}\right|=\left|\begin{array}{ccc}2&3+5\cdot2&0\\0&0+5\cdot0&1\\4&3+5\cdot4&4\end{array}\right|\)

  5. \(\det(\tilde A)=0\) if column vectors are linearly dependent,
    example:\(\;\;\left|\begin{array}{ccc}1&2&1\\ 2&4&1\\ 3&6&1\end{array}\right|=0\)

  6. \(\det(\tilde A)=\det(\tilde{A}^\top)\), hence all rules are equivalent for lines.
    Examples: \begin{eqnarray*}D&=&\left|\begin{array}{cccc}2&9&9&4 \\ 2&-3&12&8 \\ 4&8&3&-5\\ 1&2&6&4\end{array}\right|\stackrel{\mbox{\rm II-2I = II}}{=}\left|\begin{array}{cccc}2&5&9&4\\ 2&-7&12&8\\ 4&0&3&-5\\ 1&0&6&4\end{array}\right|\stackrel{\mbox{\rm III/3}}{=}3\cdot\left|\begin{array}{cccc}2&5&3&4\\ 2&-7&4&8\\ 4&0&1&-5\\ 1&0&2&4\end{array}\right|\\ \\ &=&3\left\{-5\left|\begin{array}{ccc}2&4&8\\ 4&1&-5\\ 1&2&4\end{array}\right|-7\left|\begin{array}{ccc}2&3&4\\ 4&1&-5\\ 1&2&4\end{array}\right|\right\}=0-21\left|\begin{array}{ccc}2&3&4\\ 4&1&-5\\ 1&2&4\end{array}\right|\\ \\&\stackrel{\mbox{\rm I-III = I}}{=}&-21\left|\begin{array}{ccc}1&1&0\\ 4&1&-5\\ 1&2&4\end{array}\right|=-21\left(\left|\begin{array}{cc}1&-5\\ 2&4\end{array}\right|-\left|\begin{array}{cc}4&-5\\ 1&4\end{array}\right|\right)=147\end{eqnarray*}

  7. other rules: \begin{eqnarray*}\det(\tilde I)&=&1\\ \\ \det(\tilde A\cdot\tilde B)&=&\det(\tilde A)\det(\tilde B)\\ \\ \mbox{in general:}\;\;\det(\tilde{A}+\tilde{B})&\neq&\det(\tilde{A})+\det(\tilde{B})!\end{eqnarray*}

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© J. Carstensen (Math for MS)