Commit f1257e50 by Jim Hefferon

### det1 seeing the end

parent 5234ae8f
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 ... ... @@ -22544,7 +22544,7 @@ octave:6> gplot z \end{ans} \subsection{Subsection Four.I.4: Determinants Exist} \begin{ans}{Four.I.4.10} \begin{ans}{Four.I.4.9} This is the permutation expansion of the determinant of a $\nbyn{2}$ matrix \begin{equation*} ... ... @@ -22582,7 +22582,7 @@ octave:6> gplot z is self-transpose). \end{ans} \begin{ans}{Four.I.4.11} \begin{ans}{Four.I.4.10} Each of these is easy to check. \begin{exparts*} \partsitem ... ... @@ -22602,7 +22602,7 @@ octave:6> gplot z \end{exparts*} \end{ans} \begin{ans}{Four.I.4.12} \begin{ans}{Four.I.4.11} \begin{exparts} \partsitem $$\sgn(\phi_1)=+1$$, $$\sgn(\phi_2)=-1$$ \partsitem $$\sgn(\phi_1)=+1$$, $$\sgn(\phi_2)=-1$$, ... ... @@ -22611,7 +22611,7 @@ octave:6> gplot z \end{exparts} \end{ans} \begin{ans}{Four.I.4.13} \begin{ans}{Four.I.4.12} To get a nonzero term in the permutation expansion we must use the $$1,2$$ entry and the $$4,3$$ entry. Having fixed on those two we must also use the $$2,1$$ entry and ... ... @@ -22629,7 +22629,7 @@ octave:6> gplot z $\rho_3\leftrightarrow\rho_4$ will produce the identity matrix. \end{ans} \begin{ans}{Four.I.4.14} \begin{ans}{Four.I.4.13} The pattern is this. \begin{center} \begin{tabular}[b]{c|ccccccc} ... ... @@ -22651,7 +22651,7 @@ octave:6> gplot z $-1$ and the $n=3$ case has a signum of $-1$. \end{ans} \begin{ans}{Four.I.4.15} \begin{ans}{Four.I.4.14} \begin{exparts} \partsitem Permutations can be viewed as one-one and onto maps $$\map{\phi}{\set{1,\ldots,n}}{\set{1,\ldots,n}}$$. ... ... @@ -22664,7 +22664,7 @@ octave:6> gplot z \end{exparts} \end{ans} \begin{ans}{Four.I.4.16} \begin{ans}{Four.I.4.15} If $\phi(i)=j$ then $\phi^{-1}(j)=i$. The result now follows on the observation that $P_{\phi}$ has a $1$ in entry $i,j$ if and only if $\phi(i)=j$, ... ... @@ -22672,7 +22672,7 @@ octave:6> gplot z $1$ in entry $j,i$ if and only if $\phi^{-1}(j)=i$, \end{ans} \begin{ans}{Four.I.4.17} \begin{ans}{Four.I.4.16} This does not say that $$m$$ is the least number of swaps to produce an identity, nor does it say that $$m$$ is the most. It instead says that ... ... @@ -22723,7 +22723,7 @@ octave:6> gplot z identity, the parity (evenness or oddness) is the same. \end{ans} \begin{ans}{Four.I.4.18} \begin{ans}{Four.I.4.17} \begin{exparts} \partsitem First, $g(\phi_1)$ is the product of the single factor $2-1$ and so $g(\phi_1)=1$.
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