Commit c5a40d89 by 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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