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| a 1 a 2 b 1 b 2 | = a 1 b 2 − a 2 b 1 {\displaystyle {\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}=a_{1}b_{2}-a_{2}b_{1}}
| a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 | = a 1 | b 2 b 3 c 2 c 3 | + a 2 | b 3 b 1 c 3 c 1 | + a 3 | b 1 b 2 c 1 c 2 | = a 1 | b 2 b 3 c 2 c 3 | − a 2 | b 1 b 3 c 1 c 3 | + a 3 | b 1 b 2 c 1 c 2 | {\displaystyle {\begin{aligned}{\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}&=a_{1}{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}+a_{2}{\begin{vmatrix}b_{3}&b_{1}\\c_{3}&c_{1}\end{vmatrix}}+a_{3}{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\\&=a_{1}{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}-a_{2}{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}+a_{3}{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\end{aligned}}}
