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--- determinant_of_a_3x3_matrix [2015/03/05 11:17] – rupert
+++ determinant_of_a_3x3_matrix [2015/03/24 10:05] (current) – rupert
@@ Line 1: / Line 1: @@
-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
+If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}}$ is a $3\times 3$ matrix, then
-\[\det A=a_{11}C_11+a_{12}C_{12}+a_{13}C_{13}.\]
+\[\det A=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}.\]
 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*}