Quod Erat Demonstrandum

2016/04/29

證 Cramer’s rule

Filed under: NSS,Pure Mathematics — johnmayhk @ 12:04 上午

$\left \{ \begin{array}{ll} a_1x+b_1y+c_1z=d_1\\a_2x+b_2y+c_2z=d_2\\a_3x+b_3y+c_3z=d_3\end{array}\right.$

$M=\left(\begin{array}{rcl}a_1& b_1& c_1\\a_2& b_2& c_2\\a_3& b_3& c_3\\\end{array}\right)$

$\left(\begin{array}{rcl}x\\y\\z\\\end{array}\right)=M^{-1}\left(\begin{array}{rcl}d_1\\d_2\\d_3\\\end{array}\right)$ ………. (1)

$M^{-1}\left(\begin{array}{rcl}a_1& b_1& c_1\\a_2& b_2& c_2\\a_3& b_3& c_3\\\end{array}\right)=\left(\begin{array}{rcl}1& 0& 0\\0& 1& 0\\0& 0& 1\\\end{array}\right)$

$M^{-1}\left(\begin{array}{rcl}b_1\\b_2\\b_3\\\end{array}\right)=\left(\begin{array}{rcl}0\\1\\0\\\end{array}\right)$ ………. (2)

$M^{-1}\left(\begin{array}{rcl}c_1\\c_2\\c_3\\\end{array}\right)=\left(\begin{array}{rcl}0\\0\\1\\\end{array}\right)$ ………. (3)

$M^{-1}\left(\begin{array}{rcl}d_1& b_1& c_1\\d_2& b_2& c_2\\d_3& b_3& c_3\\\end{array}\right)=\left(\begin{array}{rcl}x& 0& 0\\y& 1& 0\\z& 0& 1\\\end{array}\right)$

$\det(M^{-1})\det\left(\begin{array}{rcl}d_1& b_1& c_1\\d_2& b_2& c_2\\d_3& b_3& c_3\\\end{array}\right)=\det\left(\begin{array}{rcl}x& 0& 0\\y& 1& 0\\z& 0& 1\\\end{array}\right)$

$x=\displaystyle\frac{\det\left(\begin{array}{rcl}d_1& b_1& c_1\\d_2& b_2& c_2\\d_3& b_3& c_3\\\end{array}\right)}{\det(M)}=\frac{\Delta_x}{\Delta}$

$y=\frac{\Delta_y}{\Delta}$$z=\frac{\Delta_z}{\Delta}$