Commit f1257e50 authored by Jim Hefferon's avatar 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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