行列式の定義と性質

\[ \det(A)=a_{11}a_{22}a_{33}-a_{11}a_{21}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{31}\]

\[\det(A)=a_{11}a_{22}-a_{12}a_{21}\]

\[ \det(A)=a_{11}a_{22}a_{33}-a_{11}a_{21}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{31}\]

\[ A=\left( \begin{array}{cccc} a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\ a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ n1 } & a_{ n2 } & \ldots & a_{ nn } \end{array} \right) \]

\[\det(A)=\sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)} \tag{1}\\ \qquad\mathrm{ただしここで}\, S_n \,\mathrm{は}\,n\,\mathrm{次の置換の集合}\]

\[ A=\left(\begin{array}{rrrr} 4 & 1 & -2 & 1\\ 3 & -3 & 2 & 5\\ 1 & -1 & 3 & -2\\ -2 & 3 & 3 & 1 \end{array}\right) \]

\[ \begin{align} \det(A) &=-36-72+8-48-60-180\\ &\quad -9-18+2-8-15-30\\ &\quad\quad+6-36-6+24-30+20\\ &\quad\quad\quad+9+27-9-18-6+4\\ &=-481 \end{align} \]

\[ \mathrm{sgn}(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)} \]

\[\sigma=\left(\begin{array}{cccc} 1 & 2 &\cdots & n\\ \sigma(1) & \sigma(2) &\cdots &\sigma(n) \\ \end{array}\right)\]

\[\sigma=\left(\begin{array}{cccc} \sigma^{-1}(1) & \sigma^{-1}(2) &\cdots &\sigma^{-1}(n) \\ 1 & 2 &\cdots & n\\ \end{array}\right)\]

\[ a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)} \]

\[ a_{\sigma^{-1}(1)1}a_{\sigma^{-1}(2)2}\cdots a_{\sigma^{-1}(n)n} \]

\[\det(A)=\sum_{\sigma^{-1} \in S_n} \mathrm{sgn}(\sigma^{-1})a_{\sigma^{-1}(1)1}a_{\sigma^{-1}(2)2}\cdots a_{\sigma^{-1}(n)n} \tag{2}\\ \qquad\text{ただしここで}\, S_n \,\text{は}\,n\,\text{次の置換の集合}\]

\[\det(A)=\sum_{\tau \in S_n} \mathrm{sgn}(\tau)a_{\tau(1)1}a_{\tau(2)2}\cdots a_{\tau(n)n} \tag{3}\\ \qquad\text{ただしここで}\, S_n \,\text{は}\,n\,\text{次の置換の集合}\]

\[ \mathrm{sgn}(\tau)a_{\tau(1)1}a_{\tau(2)2}\cdots a_{\tau(n)n} \]

\[ A = \left( \begin{array}{cccc} a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\ a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ n1 } & a_{ n2 } & \ldots & a_{ nn } \end{array} \right)\]

\[\begin{align} A &=\left(\boldsymbol{a}_1\; \boldsymbol{a}_2\;\ldots\; \boldsymbol{a}_n\right)\\[6pt] &\text{ただしここで}\quad \boldsymbol{a}_1 = \left(\begin{array}{c}a_{11}\\ a_{21}\\ \vdots\\ a_{n1}\end{array}\right),\, \boldsymbol{a}_2 = \left(\begin{array}{c}a_{12}\\ a_{22}\\ \vdots\\ a_{n2}\end{array}\right),\cdots,\, \boldsymbol{a}_n = \left(\begin{array}{c}a_{1n}\\ a_{2n}\\ \vdots\\ a_{nn}\end{array}\right) \end{align}\]

\[\boldsymbol{a}_1 = \left(\begin{array}{c}a_{11}\\ a_{21}\\ \vdots\\ a_{n1}\end{array}\right), \boldsymbol{a}_2 = \left(\begin{array}{c}a_{12}\\ a_{22}\\ \vdots\\ a_{n2}\end{array}\right),\cdots, \boldsymbol{a}_n = \left(\begin{array}{c}a_{1n}\\ a_{2n}\\ \vdots\\ a_{nn}\end{array}\right),\: \boldsymbol{a}'_j = \left(\begin{array}{c}a'_{1j}\\ a'_{2j}\\ \vdots\\ a'_{nj}\end{array}\right), \boldsymbol{a}''_j = \left(\begin{array}{c}a''_{1j}\\ a''_{2j}\\ \vdots\\ a''_{nj}\end{array}\right)\]

\[\begin{align} \det(\boldsymbol{a}_{\tau(1)},\boldsymbol{a}_{\tau(2)},\cdots,\boldsymbol{a}_{\tau(n))})&=-\mathrm{sgn}(\tau)\det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) \end{align}\]

\[ \tau=\left(\begin{array}{c} 1 & 2 & \cdots & n\\ \tau(1) & \tau(2) & \cdots & \tau(n) \end{array}\right) =\left(\begin{array}{c} 1 & 2 & \cdots & n\\ k_1 & k_2 & \cdots & k_n \end{array}\right) \]

\[\det(\boldsymbol{a}_{\tau(1)},\cdots,\boldsymbol{a}_{\tau(n))}) =\sum_{\rho \in S_n}\mathrm{sgn}(\rho)a_{\rho(1)k_1}a_{\rho(2)k_2}\cdots a_{\rho(n)k_n} \tag{4}\]

\[\det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)=\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{\sigma(1)1}a_{\sigma(2)2}\cdots a_{\sigma(n)n} \tag{5}\]

\[\sigma=\left(\begin{array}{c} k_1 &k_2 & \cdots & k_n\\ \sigma(k_1) & \sigma(k_2) & \cdots & \sigma(k_n) \end{array}\right) \]

\[ \mathrm{sgn}(\sigma)a_{\sigma(1)1}a_{\sigma(2)2}\cdots a_{\sigma(n)n}\]

\[ \mathrm{sgn}(\sigma)a_{\sigma(k_1)k_1}a_{\sigma(k_2)k_2}\cdots a_{\sigma(k_n)k_n} \]

\[k_1=\tau(1),k_2=\tau(2),\cdots,k_n=\tau(n)\]

\[\begin{align} \det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) &=\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{\sigma(k_1)k_1}a_{\sigma(k_2)k_2}\cdots a_{\sigma(k_n)k_n}\\ &=\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{\sigma(\tau(1))k_1}a_{\sigma(\tau(2))k_2}\cdots a_{\sigma(\tau(n))k_n}\\ &=\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{\sigma\tau(1)k_1}a_{\sigma\tau(2)k_2}\cdots a_{\sigma\tau(n)k_n}\tag{6} \end{align}\]

\[\mathrm{sgn}(\tau)\mathrm{sgn}(\tau)=1,\mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)=\mathrm{sgn}(\sigma\tau)\]

\[\begin{align} \det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) &=\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{\sigma\tau(1)k_1}a_{\sigma\tau(2)k_2}\cdots a_{\sigma\tau(n)k_n}\\ &=\mathrm{sgn}(\tau)\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)a_{\sigma\tau(1)k_1}a_{\sigma\tau(2)k_2}\cdots a_{\sigma\tau(n)k_n}\\ &=\mathrm{sgn}(\tau)\sum_{\sigma \in S_n} \mathrm{sgn}(\sigma\tau)a_{\sigma\tau(1)k_1}a_{\sigma\tau(2)k_2}\cdots a_{\sigma\tau(n)k_n} \tag{7} \end{align}\]

\[\begin{align} \det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) &=\mathrm{sgn}(\tau)\sum_{\rho \in S_n} \mathrm{sgn}(\rho)a_{\rho(1)k_1}a_{\rho(2)k_2}\cdots a_{\rho(n)k_n} \tag{8} \end{align}\]

\[\begin{align} \det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i + r\boldsymbol{a}_j\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)&=\det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n) \end{align}\]

\[\begin{align} \det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i + r\boldsymbol{a}_j\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n) &=\det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i \cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n) +\det(\boldsymbol{a}_1,\cdots, r\boldsymbol{a}_j\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)\\ &=\det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i \cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n) +r\det(\boldsymbol{a}_1,\cdots, \boldsymbol{a}_j\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)\\ &=\det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)+0\\ &=\det(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_i\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)\\ \end{align}\]

\[\boldsymbol{a}_1 = \left(\begin{array}{c}a_{11}\\ a_{21}\\ \vdots\\ a_{n1}\end{array}\right), \boldsymbol{a}_2 = \left(\begin{array}{c}a_{12}\\ a_{22}\\ \vdots\\ a_{n2}\end{array}\right),\cdots, \boldsymbol{a}_n = \left(\begin{array}{c}a_{1n}\\ a_{2n}\\ \vdots\\ a_{nn}\end{array}\right)\]

\[\begin{align} F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}'_j+\boldsymbol{a}''_j,\cdots,\boldsymbol{a}_n)&=F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}'_j,\cdots,\boldsymbol{a}_n) +F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}''_j,\cdots,\boldsymbol{a}_n)\quad(j=1,2,\cdots,n)\\ F(\boldsymbol{a}_1,\cdots,r\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)&=rF(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)\quad(j=1,2,\cdots,n) \end{align}\]

\[\begin{align} F(\boldsymbol{a}_{\sigma(1)},\boldsymbol{a}_{\sigma(2)},\cdots,\boldsymbol{a}_{\sigma(n))})&=\mathrm{sgn}(\sigma)F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) \end{align}\]

\[\begin{align} \boldsymbol{a}_i&=a_{1i}\boldsymbol{e}_1+a_{2i}\boldsymbol{e}_2+\cdots+a_{ni}\boldsymbol{e}_n\\ &=\sum_{k_i=1}^na_{k_ii}\boldsymbol{e}_k \end{align}\]

\[\begin{align} F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) &=F\left(\sum_{k_1=1}^na_{k_11}\boldsymbol{e}_{k_1},\sum_{k_2=1}^na_{k_22}\boldsymbol{e}_{k_2},\cdots,\sum_{k_n=1}^na_{k_n n}\boldsymbol{e}_{k_n}\right)\\[6pt] &=\sum_{k_1=1}^n a_{k_1 1}F\left(\boldsymbol{e}_{k_1},\sum_{k_2=1}^na_{k_22}\boldsymbol{e}_{k_2},\cdots,\sum_{k_n=1}^na_{k_n n}\boldsymbol{e}_{k_n}\right)\\[6pt] &=\sum_{k_1=1}^n a_{k_1 1}\sum_{k_2=1}^n a_{k_2 2}F\left(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\sum_{k_n=1}^na_{k_n n}\boldsymbol{e}_{k_n}\right)\\[6pt] &= \qquad \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ &=\sum_{k_1=1}^n a_{k_1 1}\sum_{k_2=1}^n a_{k_2 2}\cdots \sum_{k_n=1}^n a_{k_n n} F\left(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\boldsymbol{e}_{k_n}\right)\\[6pt] &=\sum_{k_1=1}^n\sum_{k_2=1}^n\cdots\sum_{k_n=1}^n a_{k_1 1}a_{k_2 2}\cdots a_{k_n n} F\left(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\boldsymbol{e}_{k_n}\right)\\[6pt] &=\text{可能な}\, k_1,k_2,\ldots,,k_n\,\text{の組すべてに対する}\\ &\qquad a_{k_1 1}a_{k_2 2}\cdots a_{k_n n}F(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\boldsymbol{e}_{k_n})\,\text{の和} \end{align}\]

\[\begin{align} &F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)\\ &=k_1,k_2,\cdots,k_n \,\text{がすべて異なるときの}\\&\qquad a_{k_1 1}a_{k_2 2}\cdots a_{k_n n}F(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\boldsymbol{e}_{k_n})\,\text{の和}\\ \end{align}\]

\[\sigma=\left(\begin{array}{c} 1 &2 & \cdots & k\\ k_1 & k_2 & \cdots & k_n \end{array}\right) \]

\[ F(\boldsymbol{e}_{k_1},\boldsymbol{e}_{k_2},\cdots,\boldsymbol{e}_{k_n})=\mathrm{sgn}(\sigma)F(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n})\]

\[\begin{align} &F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)\\[6pt] &=\sum_{\sigma \in S_n}a_{\sigma(1)1}a_{\sigma(2)2}\cdots a_{\sigma(n)n}\mathrm{sgn}(\sigma)F(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n})\\[6pt] &=\left\{\sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)a_{\sigma(1)1}a_{\sigma(2)2}\cdots a_{\sigma(n)n}\right\}F(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n})\\[6pt] &=\det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)F(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\cdots,\boldsymbol{e}_{n}) \end{align}\]

\[\begin{align} F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}'_j+\boldsymbol{a}''_j,\cdots,\boldsymbol{a}_n)&=F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}'_j,\cdots,\boldsymbol{a}_n) +F(\boldsymbol{a}_1,\cdots,\boldsymbol{a}''_j,\cdots,\boldsymbol{a}_n)\quad(j=1,2,\cdots,n)\\ F(\boldsymbol{a}_1,\cdots,r\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)&=rF(\boldsymbol{a}_1,\cdots,\boldsymbol{a}_j,\cdots,\boldsymbol{a}_n)\quad(j=1,2,\cdots,n) \end{align}\]

\[\begin{align} F(\boldsymbol{a}_{\sigma(1)},\boldsymbol{a}_{\sigma(2)},\cdots,\boldsymbol{a}_{\sigma(n))})&=\mathrm{sgn}(\sigma)F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) \end{align}\]

\[F(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) =c\det(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n)\]

\[\det(AB)=\det(A)\det(B)\]

\[\begin{align} AB &=(\boldsymbol{a}_1,\boldsymbol{a}_2,\cdots,\boldsymbol{a}_n) \left( \begin{array}{cccc} b_{ 11 } & b_{ 12 } & \ldots & b_{ 1n } \\ b_{ 21 } & b_{ 22 } & \ldots & b_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ b_{ n1 } & b_{ n2 } & \ldots & b_{ nn } \end{array} \right)\\[6pt] &=\left(\sum_{k_1=1}^n\boldsymbol{a}_{k_1}b_{k_1 1},\sum_{k_2=1}^n\boldsymbol{a}_{k_2}b_{k_2 2},\cdots,\sum_{k_n=1}^n\boldsymbol{a}_{k_n}b_{k_n n}\right) \end{align}\]

\[\begin{align}\det(AB) &=\det\left(\sum_{k_1}\boldsymbol{a}_{k_1}b_{k_1 1},\sum_{k_2}\boldsymbol{a}_{k_2}b_{k_2 2},\cdots,\sum_{k_n}\boldsymbol{a}_{k_n}b_{k_n n}\right)\\[6pt] &=\text{可能な}\, k_1,k_2,\ldots,,k_n\,\text{の組すべてに対する}\\[6pt] &\qquad \det(\boldsymbol{a}_{k_1},\boldsymbol{a}_{k_2},\cdots,\boldsymbol{a}_{k_n})b_{k_1 1}b_{k_2 2}\cdots b_{k_n n}\,\text{の和}\\[6pt] &=k_1,k_2,\cdots,k_n \,\text{がすべて異なるときの}\\[6pt] &\qquad \det(\boldsymbol{a}_{k_1},\boldsymbol{a}_{k_2},\cdots,\boldsymbol{a}_{k_n})b_{k_1 1}b_{k_2 2}\cdots b_{k_n n}\,\text{の和}\\[6pt] &= \sum_{\sigma \in S_n}\det(\boldsymbol{a}_{\sigma(1)},\boldsymbol{a}_{\sigma(2)},\cdots,\boldsymbol{a}_{\sigma(n)})b_{\sigma(1) 1}b_{\sigma(2) 2}\cdots b_{\sigma(n) n}\\ &= \sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)\det(\boldsymbol{a}_{1},\boldsymbol{a}_{2},\cdots,\boldsymbol{a}_{n})b_{\sigma(1) 1}b_{\sigma(2) 2}\cdots b_{\sigma(n) n}\\ &= \det(\boldsymbol{a}_{1},\boldsymbol{a}_{2},\cdots,\boldsymbol{a}_{n})\sum_{\sigma \in S_n}\mathrm{sgn}(\sigma)b_{\sigma(1) 1}b_{\sigma(2) 2}\cdots b_{\sigma(n) n}\\ &= \det(\boldsymbol{a}_{1},\boldsymbol{a}_{2},\cdots,\boldsymbol{a}_{n})\det(B)\\[6pt] &=\det(A)\det(B) \end{align}\]