行列式の計算例

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

\[ \left|\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right| =a_{11}a_{22}-a_{12}a_{21} \]

\[ \begin{align} \left|\begin{array}{ccc} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{array}\right| &=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{31}\\ &\qquad - a_{11}a_{23}a_{31}+a_{12}a_{21}a_{33}+a_{13}a_{22}a_{31} \end{align} \]

\[ \left|\begin{array}{cccc} a_{11} & 0 & \cdots & 0\\ 0 & a_{22} & \cdots & 0\\ \vdots & \vdots &\ddots & \vdots\\ 0 & 0 & \cdots & a_{nn}\\ \end{array}\right| =a_{11}a_{22} \cdots a_{nn} \]

\[ \left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ 0 & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots &\ddots & \vdots\\ 0 & 0 & \cdots & a_{nn}\\ \end{array}\right| =a_{11}a_{22} \cdots a_{nn} \]

\[ \left| \begin{array}{ccccc} a_{ 11 } & \ldots & 0 & \ldots & a_{ 1n } \\ a_{ 21 } & \ldots & 0 & \ldots & a_{ 2n } \\ \vdots & & \vdots & & \vdots \\ a_{ n1 } & \ldots & 0 & \ldots & a_{ nn } \end{array} \right|=0 \]

\[ \left| \begin{array}{cccc} ca_{ 11 } & ca_{ 12 } & \ldots & ca_{ 1n } \\ ca_{ 21 } & ca_{ 22 } & \ldots & ca_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ ca_{ n1 } & ca_{ n2 } & \ldots & ca_{ nn } \end{array}\right| = c^n \left| \begin{array}{cccc} a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\ a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ n1 } & a_{ n2 } & \ldots & a_{ nn } \end{array} \right| \]

\[ \begin{align} A&=\left(\begin{array}{cc} A_{11} & O\\ A_{21} & A_{22} \end{array}\right)\\[6pt] &\quad \text{ただし、ここで} \,A_{11}\, \text{と} \,A_{22}\, \text{は}\\ &\quad \text{それぞれある次数の正方行列} \end{align} \]

\[ |A|=|A_{11}||A_{22}| \]

\[|A|=\left| \begin{array}{cccccc} a_{ 11 } &\ldots & a_{ 1m } & 0 &\ldots & 0 \\ \vdots & & \vdots & \vdots & & 0\\ a_{ m1 } & \ldots & a_{mm} &0 &\ldots & 0 \\ a_{ m+1\,1 } & \ldots & a_{m+1\,m} &a_{m+1\,m+1} &\ldots & a_{m+1\,m+n} \\ \vdots & \vdots & \ddots & \vdots & & \vdots \\ a_{ m+n\,1 } & \ldots & a_{m+n\,m} & a_{m+n\, m+1} &\ldots & a_{ m+n\,m+n } \end{array} \right|,\,\\[20pt] |A_{12}|=\left| \begin{array}{ccc} a_{ 11 } &\ldots & a_{ 1m } \\ \vdots & & \vdots \\ a_{ m1 } & \ldots & a_{mm} \\ \end{array} \right|,\quad |A_{22}|=\left| \begin{array}{ccc} a_{m+1\,m+1} &\ldots & a_{m+1\,m+n} \\ \vdots & & \vdots \\ a_{m+n\, m+1} &\ldots & a_{ m+n\,m+n } \end{array} \right|\]

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

\[ \mathrm{sgn}(\tau)\mathrm{sgn}(\rho)\left(a_{1\tau(1)}\cdots a_{m\tau(m)}\cdot a_{m+1\,\rho(m+1)}\ldots a_{m+n\,\rho(m+n)}\right) \]

\[ \begin{align} |A|&=\sum_{(\tau,\rho)}\left\{\mathrm{sgn}(\tau)\mathrm{sgn}(\rho)\left(a_{1\tau(1)}\cdots a_{m\tau(m)}\cdot a_{m+1\,\rho(m+1)}\ldots a_{m+n\,\rho(m+n)}\right)\right\}\\ &=\sum_{\tau}\sum_{\rho}\left\{\left(\mathrm{sgn}(\tau)a_{1\tau(1)}\cdots a_{m\tau(m)}\right)\cdot\left(\mathrm{sgn}(\rho) a_{m+1\,\rho(m+1)}\ldots a_{m+n\,\rho(m+n)}\right)\right\}\\ &=\sum_{\tau}\mathrm{sgn}(\tau)a_{1\tau(1)}\cdots a_{m\tau(m)}\cdot\sum_{\rho}\mathrm{sgn}(\rho) a_{m+1\,\rho(m+1)}\ldots a_{m+n\,\rho(m+n)}\\ &=|A_{11}||A_{22}| \end{align} \]

\[ \begin{align} \left|\begin{array}{rrrr} -2 & 2 & 1& 2 \\ 1 & 0 & 4 & -4 \\ 3 & -5 & 2 & -9 \\ 4 & 1& -7 & 3 \end{array} \right| &\stackrel{3\text{列}+4\text{列}\times 1\,}{=} \left|\begin{array}{rrrr} -2 & 2 & 3 & 2 \\ 1 & 0 & 0 & -4 \\ 3 & -5 & -7 & -9 \\ 4 & 1& -4 & 3 \end{array} \right|\\[15pt] &\stackrel{4\text{列}+1\text{列}\times 4\,}{=} \left|\begin{array}{rrrr} -2 & 2 & 3& -6 \\ 1 & 0 & 0 & 0 \\ 3 & -5 & -7 & 3 \\ 4 & 1& -4 & 19 \end{array} \right|\\[15pt] &\stackrel{\text{第}\,2\,\text{行で展開}}{=} 1\times(-1)^{2+1}\left|\begin{array}{rrr} 2 & 3& -6 \\ -5 & -7 & 3 \\ 1& -4 & 19 \end{array} \right| + 0\times (-1)^{2+2}\left|\begin{array}{rrr} * & * & *\\ * & * & * \\ * & * & *\\ \end{array} \right|\\[15pt] & \qquad \qquad \qquad+ 0 \times (-1)^{2+3} \left|\begin{array}{rrr} * & * & *\\ * & * & * \\ * & * & *\\ \end{array} \right| + 0\times (-1)^{2+4} \left|\begin{array}{rrr} * & * & *\\ * & * & * \\ * & * & *\\ \end{array} \right|\\[15pt] &\quad=\quad (-1)\times \left|\begin{array}{rrr} 2 & 3& -6 \\ -5 & -7 & 3 \\ 1& -4 & 19 \end{array} \right|\\[15pt] &\stackrel{1\text{行}-3\text{行}\times 2\,}{=}(-1)\times \left|\begin{array}{rrr} 0 & 11 & -44 \\ -5 & -7 & 3 \\ 1& -4 & 19 \end{array} \right|\\ &\stackrel{2\text{行}+3\text{行}\times 5\,}{=}(-1)\times \left|\begin{array}{rrr} 0 & 11 & -44 \\ 0 & -27 & 98 \\ 1& -4 & 19\\ \end{array} \right|\\[15pt] &\stackrel{\text{第}\,1\,\text{列で展開}}{=} (-1)\times\left\{0\times(-1)^{1+1}\times \left|\begin{array}{rr} * & * \\ * & * \end{array} \right|\right.\\[15pt] &\qquad\qquad\qquad\qquad\qquad\qquad\left.+0\times (-1)^{2+1}\times \left|\begin{array}{rr} * & * \\ * & * \end{array} \right|\right.\\[15pt] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+1\times (-1)^{3+1}\times \left|\begin{array}{rr} 11 & -44 \\ -27 & 98 \\ \end{array} \right|\right\}\\[15pt] &\quad = \quad -1\times\left\{11\times74-(-44)\times(-9)\right\}=-1\times (-110) =110 \end{align} \]