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Problems PreCal 1508 PLTL Workshop, October 19, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

\end{center}
Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}

\begin{enumerate}
\item Perform partial fraction decomposition.
\[ \frac{x^2 + 2x + 3}{x^3 + x}\]

\item Perform the following operations on the matricies (if possible)
\[
A = \begin{bmatrix}
1 & 0 & 2 \\ 
2 & 1 & 6 \\ 
-4 & 2 & -3
\end{bmatrix} 
\quad\quad\quad
B = \begin{bmatrix}
2 & 1 \\ 
1 & 4 \\ 
0 & 3
\end{bmatrix} 
\]
\begin{center}
\begin{inparaenum}[a)]
\item $2A + B\quad\quad$
\item $AB\quad\quad$
\item $BA\quad\quad$
\item $A^2$
\end{inparaenum}
\end{center}

\item Solve the following system of equations using Gau\ss-Jordan elimination

\[ \left\{\begin{array}{lr}
	-x + y  -z = -5\\
	 4y + 2z = 0\\
	 x + 2y - 3z = -28\\
	\end{array}
	\right.\quad\quad\quad
\]

\item Perform the following operations on the matricies (vectors).
\[
A = \begin{pmatrix}
1 & 2 & 3
\end{pmatrix} \quad\quad\quad\quad
B = \begin{pmatrix}
3 \\ 
2 \\ 
1
\end{pmatrix}\]

\begin{center}
\begin{inparaenum}[a)]
	\item  $A\cdot B\quad\quad\quad$
	\item  $B\cdot A$
\end{inparaenum}
\end{center}

\item Are the following true or false? Explain/give a counterexample.
\begin{enumerate}
	\item For all matricies $A\cdot B = B\cdot A$
	\item For all matricies $A + B = B + A$
	\item For all matricies $A \cdot I = I\cdot A = A$. Where $I$ is the Identity matrix (1's on the diagonal, 0's elsewhere)
\end{enumerate}

\item For which class of matrices, $A$ is $A\cdot B = B \cdot A$ (for any $B$) true?

\begin{notation}
Linear systems are often written as $A\cdot x =b$ where $b$ is a known vector, and $x$ is a vector of unknowns to be solved for, and the elements of $A$ are the coefficients of $x_1, x_2, ...$.
\end{notation}
\item Write the system of equations presented in problem 3 in this form. How could you solve this if you knew $A^{-1}$?
\end{enumerate}
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