行列式の計算

(2)
$$\begin{array}{rcl}\overline{)\begin{array}{cccc}1& 1& 2& 1\\ 3& 1& 4& 3\\ 7& 6& 1& 2\\ 1& 1& 3& 4\end{array}}& =& \overline{)\begin{array}{cccc}1& 0& 0& 0\\ 3& -2& -2& 0\\ 7& -1& -13& -5\\ 1& 0& 1& 3\end{array}}\\ & =& \overline{)\begin{array}{ccc}-2& -2& 0\\ -1& -13& -5\\ 0& 1& 3\end{array}}\\ & =& \overline{)\begin{array}{ccc}-2& 0& 0\\ -1& –12& -5\\ 0& 1& 3\end{array}}\\ & =& -2\times \overline{)\begin{array}{cc}-12& -5\\ 1& 3\end{array}}\\ & =& 62\end{array}$$

(3)
$$\begin{array}{rcl}\overline{)\begin{array}{cccc}1& 1& 1& 1\\ r& 1& 1& 1\\ r& r& 1& 1\\ r& r& r& 1\end{array}}& =& \overline{)\begin{array}{cccc}0& 0& 0& 1\\ r–1& 0& 0& 1\\ r–1& r–1& 0& 1\\ r–1& r–1& r–1& 1\end{array}}\\ & =& -(r–1)\times \overline{)\begin{array}{cc}r–1& 0\\ r–1& r–1\end{array}}\\ & =& –(r–1{)}^3\end{array}$$

(4)
$$\begin{array}{rcl}\overline{)\begin{array}{ccc}2a& 2b& b–c\\ 2b& 2a& a+c\\ a+b& a+b& b\end{array}}& =& \overline{)\begin{array}{ccc}2a& 2(b–a)& b–c\\ 2b& 2(a–b)& a+c\\ a+b& 0& b\end{array}}\\ & =& \overline{)\begin{array}{ccc}2(a+b)& 0& a+b\\ 2b& 2(a–b)& a+c\\ a+b& 0& b\end{array}}\\ & =& \overline{)\begin{array}{cc}2(a+b)& a+b\\ a+b& b\end{array}}\\ & =& (a+b)\times \overline{)\begin{array}{cc}2& 1\\ a+b& b\end{array}}\\ & =& –(a–b{)}^2\end{array}$$