HackMD - Collaborative Markdown Knowledge Base

### Draft for calculating determinant **Warning: This is a draft! Has a lot of errors and typos** Resume at 5:06 https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/ryFZb1w2L $$ \newcommand\fy{-1} \newcommand\r[1]{{\color{red}#1}} $$ $$ \det(M)=\begin{pmatrix} 1&1&\fy&1\\ 0&1&\fy&0\\ \fy&1&\fy&0\\ 1&1&\fy&1 \end{pmatrix} $$ Multiply the thrid column by $-1$ $$ \det\begin{pmatrix} 1&1&\fy&1\\ 0&1&\fy&0\\ \fy&1&\fy&0\\ 1&1&\fy&1 \end{pmatrix} = - \det\begin{pmatrix} 1&1&1&1\\ 0&1&1&0\\ \fy&1&1&0\\ 1&1&1&1 \end{pmatrix}=0 $$ **Another method** $$ \det\begin{pmatrix} 1&1&\fy&1\\ 0&1&\fy&0\\ \fy&1&\fy&0\\ 1&1&\fy&1 \end{pmatrix} = \det\begin{pmatrix} 1&1&\fy&1\\ 0&1&\fy&0\\ 0&2&-2&1\\ 0&0&0&0 \end{pmatrix} $$$$=1\det\begin{pmatrix} 1&-1&0\\2&-2&1\\0&0&0 \end{pmatrix} $$ See all possible terms corresponding to partition? $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&\r1&\fy&0\\ \fy&1&\r\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ -1 $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&\r1&\fy&0\\ \fy&1&\fy&\r0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&1&\r\fy&0\\ \fy&\r1&\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ -1 (odd permutation) count as 1 $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&1&\r\fy&0\\ \fy&1&\fy&\r0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&1&\fy&\r0\\ \fy&\r1&\fy&0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} \r 1&1&\fy&1\\ 0&1&\fy&\r0\\ \fy&1&\r\fy&0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1&\r 1&\fy&1\\ \r0&1&\fy&0\\ \fy&1&\r\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1&\r 1&\fy&1\\ \r0&1&\fy&0\\ \fy&1&\fy&\r0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1&\r 1&\fy&1\\ 0&1&\r\fy&0\\ \fy&1&\fy&\r0\\ \r1&1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1&\r 1&\fy&1\\ 0&1&\r\fy&0\\ \r\fy&1&\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ 1(even permutation) $$ \begin{pmatrix} 1&\r 1&\fy&1\\ 0&1&\fy&\r0\\ \r\fy&1&\fy&0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1&\r 1&\fy&1\\ 0&1&\fy&\r0\\ \fy&1&\r\fy&0\\ \r1&1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\r\fy&1\\ \r0&1&\fy&0\\ \fy&\r1&\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\r\fy&1\\ \r0&1&\fy&0\\ \fy&1&\fy&\r0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\r\fy&1\\ 0&\r1&\fy&0\\ \r\fy&1&\fy&0\\ 1&1&\fy&\r1 \end{pmatrix} $$ 1(odd permutation) count as $-1$ $$ \begin{pmatrix} 1& 1&\r\fy&1\\ 0&\r1&\fy&0\\ \fy&1&\fy&\r0\\ \r1&1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\r\fy&1\\ 0&1&\fy&\r0\\ \r\fy&1&\fy&0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\r\fy&1\\ 0&1&\fy&\r0\\ \fy&\r1&\fy&0\\ \r1&1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\fy&\r1\\ \r0&1&\fy&0\\ \fy&\r1&\fy&0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\fy&\r1\\ \r0&1&\fy&0\\ \fy&1&\r\fy&0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 0 $$ \begin{pmatrix} 1& 1&\fy&\r1\\ 0&\r1&\fy&0\\ \r\fy&1&\fy&0\\ 1&1&\r\fy&1 \end{pmatrix} $$ 1(even permutation) $$ \begin{pmatrix} 1& 1&\fy&\r1\\ 0&\r1&\fy&0\\ \fy&1&\r\fy&0\\ \r1&1&\fy&1 \end{pmatrix} $$ (-1)(odd permutation) count as 1 $$ \begin{pmatrix} 1& 1&\fy&\r1\\ 0&1&\r\fy&0\\ \r\fy&1&\fy&0\\ 1&\r1&\fy&1 \end{pmatrix} $$ 1 (odd permutation) count as -1 $$ \begin{pmatrix} 1& 1&\fy&\r1\\ 0&1&\r\fy&0\\ \fy&\r1&\fy&0\\ \r1&1&\fy&1 \end{pmatrix} $$ -1(even permutation)