This is an old revision of the document!
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/action.php on line 14
If $A=\mat{a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}}$ is a $3\times 3$ matrix, then \[\det A=a_{11}C_11+a_{12}C_{12}+a_{13}C_{13}.\] Here $C_{ij}$ are the cofactors of $A$.
This formula is called the Laplace expansion of $\det A$ along the first row, since $a_{11}$, $a_{12}$ and $a_{13}$ make up the first row of $A$.
Example
\begin{align*}\det\mat{1&2&3\\7&8&9\\11&12&13} &= 1\cdot C_{11} + 2 C_{12} + 3 C_{13}\\ &= 1 \cdot (+M_{11}) + 2 \cdot (-M_{12}) + 3 \cdot(+M_{13})\\ &= M_{11}-2M_{12}+3M_{13}\\ &= \det\mat{8&9\\12&13} -2\det\mat{7&9\\11&13} + 3\det\mat{7&8\\11&12}\\ &= (8\cdot 13-9\cdot 12) -2(7\cdot 13-9\cdot 11)+3(7\cdot 12-8\cdot 11)\\ &=-4 -2(-8)+3(-4)\\ &=-4+16-12\\ &=0.\end{align*}