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\begin{center} 
{\Large TEST 3}

Math 105 - 5
\end{center}

Show all your work and make your work legible.

\begin{enumerate}

\item  Name \rule{5in}{.3pt}

\item  (12 points)  Find the following if possible.  If not possible, say so.

\[
A=\left[ \begin{array}{rrr}
3 & 2 & 2 \\
2 & 2 & 2 \\
-4 & 4 & 3 \\
\end{array} \right]\hspace{.5in}
B= \left[ \begin{array}{rrr}
2 & -1 & 0 \\
3 & 4 & -2 \\
\end{array} \right]
\]
\begin{enumerate}
\item  $AB$
\vspace{1.5in}
\item  $BA$
\vspace{1.5in}
\item  $A^{-1}$
\vspace{1.5in}
\item  $B^{-1}$
%\vspace{1.5in}
\end{enumerate}
\newpage

\item  (10 points)  Find the inverse of the matrix $A$
\[
A=\left[ \begin{array}{rr}
2 & 3 \\
1 & 5 \\
\end{array} \right]
\]
\vspace{2.5in}

\item   (10 points)  Find the area of the triangle with vertices at: $(0,-2),(-1,4),(3,5)$
\vspace{2.5in}

\item  (10 points)  Use Cramer's Rule to solve the system of equations:
\[
\begin{array}{rrrrr}
2x & - & 3y & = & 7 \\
4x & - & 5y & = & 3 \\
\end{array}
\]
%\vspace{2.5in}
\newpage

\item  (8 points)  Find the sum
\[
\sum_{n=0}^{4}n^2
\]
\vspace{2.5in}

\item (8 points)  Find the sum
\[
\sum_{j=2}^{34}2j+5
\]
\vspace{2.2in}

\item  (8 points)  Find the sum
\[
\sum_{k=1}^{10}3(-2)^{k-1}
\]
%\vspace{2.2in}
\newpage

\item  (10 points)  Find the common ratio $r$,and $a_1$ of a geometric sequence if $a_3=75$ and $a_5=3$.
%\newpage
\vspace{2.in}

\item  (10 points)  Find $_{12}C_5$
\vspace{2.in}

\item  (15 points)  Use mathematical Induction to prove the following:
\[
1+3+5+\ldots +(2n-1) = n^2
\]
%\vspace{5.5in}




\end{enumerate}

\end{document}





