
\subsection{\bf Equations for Eigenvalues---II}

\begin{slide}

\begin{itemize}
\item Condition for nontrivial $\underline{v}$ to solve
$B\underline{v} = 0$ is:\ \ determinant $(B)=0$

\item This corresponds to
\begin{eqnarray*}
\bullet\ n=2 \hspace{3em} \mbox{det}\ B & = & b_{11}\, b_{22} -
b_{12} b_{21} \\
\bullet\ n=3 \hspace{3em} \mbox{det}\ B & = & b_{11}\, \mbox{det}\
\left\lbrack \begin{array}{cc}
b_{22} & b_{23} \\
b_{32} & b_{33} \end{array} \right\rbrack \\
& - & b_{12}\, \mbox{det}\ \left\lbrack \begin{array}{cc}
b_{21} & b_{23} \\
b_{31} & b_{33} \end{array} \right\rbrack \\
& + & b_{13}\, \mbox{det}\ \left\lbrack \begin{array}{cc}
b_{22} & b_{23} \\
b_{32} & b_{33} \end{array} \right\rbrack \\
\bullet \mbox{and\ so\ on\ recursively}
\end{eqnarray*}
\end{itemize}

\end{slide}

