Man berechne
|4−2−83151812|=4⋅1⋅12−2⋅5⋅1−8⋅3⋅8+8⋅1⋅1+2⋅3⋅12−4⋅5⋅8=−234{\displaystyle {\begin{vmatrix}4&-2&-8\\3&1&5\\1&8&12\end{vmatrix}}=4\cdot 1\cdot 12-2\cdot 5\cdot 1-8\cdot 3\cdot 8+8\cdot 1\cdot 1+2\cdot 3\cdot 12-4\cdot 5\cdot 8=-234}
|41−83351−412|=4⋅3⋅12+1⋅5⋅1+8⋅3⋅4+8⋅3⋅1−1⋅3⋅12+4⋅5⋅4=313{\displaystyle {\begin{vmatrix}4&1&-8\\3&3&5\\1&-4&12\end{vmatrix}}=4\cdot 3\cdot 12+1\cdot 5\cdot 1+8\cdot 3\cdot 4+8\cdot 3\cdot 1-1\cdot 3\cdot 12+4\cdot 5\cdot 4=313}
|41−23311−48|=4⋅3⋅8+1⋅1⋅1+2⋅3⋅4+2⋅3⋅1−1⋅3⋅8+4⋅1⋅4=119{\displaystyle {\begin{vmatrix}4&1&-2\\3&3&1\\1&-4&8\end{vmatrix}}=4\cdot 3\cdot 8+1\cdot 1\cdot 1+2\cdot 3\cdot 4+2\cdot 3\cdot 1-1\cdot 3\cdot 8+4\cdot 1\cdot 4=119}
|0103741−2−833151−4812|=0⋅|1−2−8315−4812|−10⋅|4−2−83151812|+3⋅|41−83351−412|−7⋅|41−23311−48|=−10⋅−234+3⋅313−7⋅119=2340+939−833=2446{\displaystyle {\begin{alignedat}{2}{\begin{vmatrix}0&10&3&7\\4&1&-2&-8\\3&3&1&5\\1&-4&8&12\end{vmatrix}}&=0\cdot {\begin{vmatrix}1&-2&-8\\3&1&5\\-4&8&12\end{vmatrix}}-10\cdot {\begin{vmatrix}4&-2&-8\\3&1&5\\1&8&12\end{vmatrix}}+3\cdot {\begin{vmatrix}4&1&-8\\3&3&5\\1&-4&12\end{vmatrix}}-7\cdot {\begin{vmatrix}4&1&-2\\3&3&1\\1&-4&8\end{vmatrix}}\\&=-10\cdot -234+3\cdot 313-7\cdot 119\\&=2340+939-833\\&=2446\end{alignedat}}}
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