ガウスの消去法
[ 1 m 12 ⋯ m 1 r − d 11 ′ ⋯ − d 1 s ′ c 1 ′ 0 1 ⋯ m 2 r − d 21 ′ ⋯ − d 2 s ′ c 2 ′ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 1 − d r 1 ′ ⋯ − d r s ′ c r ′ 0 0 ⋯ 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 ⋯ 0 0 ⋯ 0 0 ] 1 x 1 + m 12 x 2 + ⋯ + m 1 r x r − d 11 ′ x r + 1 − ⋯ − d 1 s ′ x n = c 1 ′ 0 x 1 + 1 x 2 + ⋯ + m 2 r x r − d 21 ′ x r + 1 − ⋯ − d 2 s ′ x n = c 2 ′ ⋮ 0 x 1 + 0 x 2 + ⋯ + 1 x r − d r 1 ′ x r + 1 − ⋯ − d r s ′ x n = c r ′ 0 x 1 + 0 x 2 + ⋯ + 0 x r + 0 x r + 1 + ⋯ + 0 x n = 0 ⋮ 0 x 1 + 0 x 2 + ⋯ + 0 x r + 0 x r + 1 + ⋯ + 0 x n = 0 {\displaystyle \left[{\begin{array}{ccccccc|c}1&m_{12}&\cdots &m_{1r}&-d'_{11}&\cdots &-d'_{1s}&c'_{1}\\0&1&\cdots &m_{2r}&-d'_{21}&\cdots &-d'_{2s}&c'_{2}\\\vdots &\vdots &\ddots &\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&-d'_{r1}&\cdots &-d'_{rs}&c'_{r}\\0&0&\cdots &0&0&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &0&0&\cdots &0&0\end{array}}\right]\qquad {\begin{aligned}&1x_{1}+m_{12}x_{2}+\cdots +m_{1r}x_{r}-d'_{11}x_{r+1}-\cdots -d'_{1s}x_{n}=c'_{1}\\&0x_{1}+1x_{2}+\cdots +m_{2r}x_{r}-d'_{21}x_{r+1}-\cdots -d'_{2s}x_{n}=c'_{2}\\&\qquad \vdots \\&0x_{1}+0x_{2}+\cdots +1x_{r}-d'_{r1}x_{r+1}-\cdots -d'_{rs}x_{n}=c'_{r}\\&0x_{1}+0x_{2}+\cdots +0x_{r}+0x_{r+1}+\cdots +0x_{n}=0\\&\qquad \vdots \\&0x_{1}+0x_{2}+\cdots +0x_{r}+0x_{r+1}+\cdots +0x_{n}=0\end{aligned}}}