Commit c5a40d89 authored by Jim Hefferon's avatar Jim Hefferon

final edits on det2

parent 8aef79a2
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......@@ -23223,7 +23223,7 @@ octave:6> gplot z
\end{exparts}
\end{ans}
\subsection{Subsection Four.III.1: Laplace's Expansion}
\subsection{Subsection Four.III.1: Laplace's Expansion Formula}
\begin{ans}{Four.III.1.13}
\begin{exparts*}
\partsitem \( (-1)^{2+3}\begin{vmatrix}
......@@ -3673,7 +3673,7 @@ matrix.
\ & &\vdots & & \\
& & &t_{i,\phi(j)} & \\
& &\vdots & & \\
& t_{j,\phi(i)} & & & \\
&t_{j,\phi(i)} & & & \\
& &\vdots & &
\end{mat}
\end{equation*}
......@@ -3766,7 +3766,7 @@ Do the algebra.
+d(T)
\end{aligned}
\end{align*}
To finish observe that the terms
Finish by observing that the terms
\( t_{1,\phi(1)}\cdots t_{i,\phi(i)} \cdots t_{i,\phi(j)}\cdots t_{n,\phi(n)}
\sgn(\phi) \) add to zero:
this sum represents \( d(S) \) where \( S \) is a matrix equal to \( T \)
......
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% Chapter 4, Section 3 _Linear Algebra_ Jim Hefferon
% http://joshua.smcvt.edu/linalg.html
% 2001-Jun-12
\section{Other Formulas}
\section{Laplace's Expansion}
\textit{(This section is optional.
Later sections do not depend on this material.)}
Determinants are a fount of interesting and amusing formulas.
Here is one that is often seen in calculus classes and used
Determinants are a font of interesting and amusing formulas.
Here is one that is often used
to compute determinants by hand.
\subsectionoptional{Laplace's Expansion}
\subsectionoptional{Laplace's Expansion Formula}
\index{Laplace expansion|(}
\begin{example} \label{ex:ExpThreeFirstRow}
In this permutation expansion
......
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