ガウスの消去法
1 x 1 + 0 x 2 + ⋯ + 0 x r − d 11 x r + 1 − ⋯ − d 1 s x n = c 1 0 x 1 + 1 x 2 + ⋯ + 0 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 [ 1 0 ⋯ 0 − d 11 ⋯ − d 1 s c 1 0 1 ⋯ 0 − 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 ] {\displaystyle {\begin{aligned}&1x_{1}+0x_{2}+\cdots +0x_{r}-d_{11}x_{r+1}-\cdots -d_{1s}x_{n}=c_{1}\\&0x_{1}+1x_{2}+\cdots +0x_{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}}\qquad \left[{\begin{array}{ccccccc|c}1&0&\cdots &0&-d_{11}&\cdots &-d_{1s}&c_{1}\\0&1&\cdots &0&-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]}