Differences

This shows you the differences between two versions of the page.

--- lecture_13_slides [2017/03/06 17:44] – [Examples of cofactors] rupert
+++ lecture_13_slides [2017/03/06 17:48] (current) – [Example] rupert
@@ Line 160: / Line 160: @@
   * = $-4 -2(-8)+3(-4)$
   * = $0$.
+  * (So this matrix isn't invertible!)
-  * $\mat{1&2&3\\7&8&9\\11&12&13}$ has zero determinant
-  * So this matrix is //not// invertible.
 ==== Notation ====
@@ Line 185: / Line 183: @@
 \begin{align*}
-\vm{\color{red}1&\color{red}0&\color{red}2&\color{red}3\\0&2&1&-1\\2&0&0&1\\3&0&4&2} &= \color{red}1\vm{\color{blue}2&\color{blue}1&\color{blue}-1\\0&0&1\\0&4&2}-\color{red}0\vm{0&1&-1\\2&0&1\\3&4&2}+\color{red}2\vm{\color{orange}0&\color{orange}2&\color{orange}{-1}\\2&0&1\\3&0&2}-\color{red}3\vm{\color{purple}0&\color{purple}2&\color{purple}1\\2&0&0\\3&0&4}\\
+\def\vm#1{\left|\begin{smallmatrix}#1\end{smallmatrix}\right|}
-&= 1\left(\color{blue}2\vm{0&1\\4&2}-\color{blue}1\vm{0&1\\0&2}\color{blue}{-1}\vm{0&0\\0&4}\right)\\&\quad +2\left(\color{orange}0-\color{orange}{2}\vm{2&1\\3&2}\color{orange}{-1}\vm{2&0\\3&0}\right)\\&\quad - 3\left(\color{purple}0-\color{purple}2\vm{2&0\\3&4}+\color{purple}1\vm{2&0\\3&0}\right)\\
+\vm{\color{red}1&\color{red}0&\color{red}2&\color{red}3\\0&2&1&-1\\2&0&0&1\\3&0&4&2}
-&=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)\\
+&= \color{red}1\vm{\color{blue}2&\color{blue}1&\color{blue}-1\\0&0&1\\0&4&2}-\color{red}0\vm{0&1&-1\\2&0&1\\3&4&2}+\color{red}2\vm{\color{orange}0&\color{orange}2&\color{orange}{-1}\\2&0&1\\3&0&2}-\color{red}3\vm{\color{purple}0&\color{purple}2&\color{purple}1\\2&0&0\\3&0&4}
-&=-8-4+48\\
+\\&= 1\left(\color{blue}2\vm{0&1\\4&2}-\color{blue}1\vm{0&1\\0&2}\color{blue}{-1}\vm{0&0\\0&4}\right)
+\\&\quad -0+2\left(\color{orange}0-\color{orange}{2}\vm{2&1\\3&2}\color{orange}{-1}\vm{2&0\\3&0}\right)
+\\&\quad -3\left(\color{purple}0-\color{purple}2\vm{2&0\\3&4}+\color{purple}1\vm{2&0\\3&0}\right)
+\\&=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)
+\\&=-8-4+48\\
 &=36.
 \end{align*}