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{\bf Question}

Evaluate the following determinants: $$\left| \begin{array}{cc} 15
& 19 \\ 13 & 16 \end{array} \right| \hspace{.5in} \left|
\begin{array}{ccc} 76 & -23 & 53 \\ 103 & 199 & 302 \\ 176 & 177 & 353
\end{array} \right|.$$

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{\bf Answer}

\begin{eqnarray*}\left| \begin{array}{cc} 15 & 19 \\ 13 & 16
\end{array} \right| & = &  \left| \begin{array}{cc}2 & 3 \\ 13 & 16
\end{array} \right| \\ & = & \left| \begin{array}{cc} 2 & 3 \\ 1 &
-2\end{array} \right| \\ & = &  -7 \end{eqnarray*}


\begin{eqnarray*} \left| \begin{array}{ccc} 76 & -23 & 53 \\ 103 & 199 & 302
\\ 176 & 177 & 353 \end{array} \right| & = &  \left| \begin{array}{ccc}
76 & -23 & 53 \\ 103 & 199 & 302 \\ 100 & 200& 300
\end{array} \right| \\ & = & \left| \begin{array}{ccc} 76 & -23 & 53 \\
3 & -1 & 2 \\ 100 & 200 & 300 \end{array} \right|
\\ & = & 100 \left| \begin{array}{ccc} 76 & -23 & 53 \\ 3 & -1 & 2 \\ 1 & 2 & 3
\end{array} \right| \\ & = & 100 \left| \begin{array}{ccc} 1 & 2 & 3 \\
3& -1 & 2 \\ 1& 2 & 3\end{array} \right|\\ & = & 0\end{eqnarray*}


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