定義:|abcd|=ad−bc{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc}
例如:|2314|=2⋅4−3⋅1=5{\displaystyle {\begin{vmatrix}2&3\\1&4\end{vmatrix}}=2\cdot 4-3\cdot 1=5}
方程式組為:{a1x+b1y=c1a2x+b2y=c2{\displaystyle {\begin{cases}a_{1}x+b_{1}y=c_{1}\\a_{2}x+b_{2}y=c_{2}\end{cases}}}
其解如下: x=|c1b1c2b2||a1b1a2b2|,y=|a1c1a2c2||a1b1a2b2|{\displaystyle x={\frac {\left|{\begin{matrix}c_{1}&b_{1}\\c_{2}&b_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}},\qquad y={\frac {\left|{\begin{matrix}a_{1}&c_{1}\\a_{2}&c_{2}\end{matrix}}\right|}{\left|{\begin{matrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{matrix}}\right|}}}
或是直接套公式:
x=c1b2−c2b1a1b2−a2b1{\displaystyle x={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}}} ,
y=a1c2−a2c1a1b2−a2b1{\displaystyle y={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}}
例:
{2x+37y=22x+119y=2{\displaystyle {\begin{cases}2x+37y=2\\2x+119y=2\end{cases}}}
解:x=2×119−2×372×119−2×37=238−74238−74{\displaystyle x={\frac {2\times 119-2\times 37}{2\times 119-2\times 37}}={\frac {238-74}{238-74}}} ,
238-74=164, x = 1
y=4−4238−74{\displaystyle y={\frac {4-4}{238-74}}}
4-4=0,238-74=164, y = 0