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\noindent
\parbox{2.25in}{\bf Math 5230}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Aug. $26^\mathrm{th}$, 2016}

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\begin{description}
\item[\#1 RREF] \quad First, (1) use Gaussian elimination (as defined on the handout) to compute the RREF of $A$. Then, (2) give a vector version of the general solution of $A\B x = \B 0$. Finally, (3) suppose $A=[C\; : \; \B b]$ ($C$ is all but the last column and $\B b$ is the last column of $A$). Give a vector form of the general solution of $C\B x = \B b$. 

$$\mbox{(a)} \ \  A=\begin{bmatrix} 1 & -2 & 0 & 2 & 3 \\ 2 & -4 & 1 & 5 & 4 \\ -1 & 2 & 1 & -1 & -5 \end{bmatrix} \qquad \qquad \qquad
    \mbox{(b)} \ \  A=\begin{bmatrix} 0 & 1 & 3 & 1 \\ 2 & 0 & 4 & 2 \\ 1 & -1 & -1 & 3 \end{bmatrix}$$

%\begin{enumerate}[(a)]
%\item $A=\begin{bmatrix} 1 & -2 & 0 & 2 & 3 \\ 2 & -4 & 1 & 5 & 4 \\ -1 & 2 & 1 & -1 & -5 \end{bmatrix}$
%\item $A=\begin{bmatrix} 0 & 1 & 3 & 1 \\ 2 & 0 & 4 & 2 \\ 1 & -1 & -1 & 3 \end{bmatrix}$
%\end{enumerate}

\item[\#2 PLU and Partial Pivoting] \quad First, (1) use partial pivoting to row reduce $A$ (to REF not RREF), and compute a PLU decomposition for $A$. Then (2) use this decomposition to solve $A\B x = \B b$.

$$\mbox{(a)} \ \  A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \mbox{\   and \   } \B b=\begin{bmatrix} 5 \\ 6 \end{bmatrix} \qquad \qquad \qquad
    \mbox{(b)} \ \  A=\begin{bmatrix} 1 & 4 & 1 & 0 \\ 2 & 3 & 0 & 1 \\ 3 & 2 & 1 & 1 \\ 4 & 1 & 0 & 1 \end{bmatrix} \mbox{\   and \   } \B b= \begin{bmatrix} 2 \\ 1 \\ 1 \\ 3 \end{bmatrix}$$

%\begin{enumerate}[(a)]
%\item $A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $\B b=\begin{bmatrix} 5 \\ 6 \end{bmatrix}$
%\item $A=\begin{bmatrix} 1 & 4 & 1 & 0 \\ 2 & 3 & 0 & 1 \\ 3 & 2 & 1 & 1 \\ 4 & 1 & 0 & 1 \end{bmatrix}$ and $\B b= \begin{bmatrix} 2 \\ 1 \\ 1 \\ 3 \end{bmatrix}$
%\end{enumerate}


\item[\#3 Matrix Mechanics] \quad First, (1) give a formula for a matrix which performs the given list of row operations. For example: Swapping rows 1 and 2 is $E=I-E_{11}-E_{22}+E_{12}+E_{21}$.  Then, (2) give a formula for its inverse (for my example $E^{-1}=E=I-E_{11}-E_{22}+E_{12}+E_{21}$). Next, (3) what happens if you multiply a matrix by your $E$ on the right? (my example swaps columns 1 and 2). Finally, (4) give a concrete example to show off this behavior (for my example I could take some $2 \times 2$ matrix and show how $E$ swaps its rows and columns when multiplying on the left or right). 

\begin{enumerate}[(a)]
   \item Find a matrix $E$ which swaps rows 3 and 4 and then scales row 2 by 5.
   \item Find a matrix $E$ which swaps rows 1 and 4 and then adds -2 times row 2 to row 3.
\end{enumerate}

\item[\#4 Matrix Mechanics II: Matrices Arise] \quad Let $B=I-E_{11}+3E_{22}-E_{33}-E_{99}+E_{39}+E_{93}$. What does $BA$ do to $A$? What does $AB$ do to $A$? (Assume that $A$ and $B$ are appropriately sized.\footnote{In this day and age is it ok to talk about ``appropriately sized'' matrices? Shouldn't we accept a matrix just the way it is no matter how big or small it is? I think Dr. Cook might be encouraging bias. I should probably {\color{blue}\underline{\href{https://bias.appstate.edu}{fill out a form}}}.})


\item[\#5 Matrix Mechanics III: Revenge of the Matrices] \quad Let $\mathbb{R}^{n \times n}$ be the collection of all square $n \times n$ matrices with real entries. We say that $A \in \mathbb{R}^{n \times n}$ is in the {\bf center} of $\mathbb{R}^{n \times n}$ if $AB=BA$ for all $B \in \mathbb{R}^{n \times n}$. In other words, central matrices commute with all matrices of the same size. 

Show that the $A$ is in the center if and only if $A=cI$ for some $c \in \mathbb{R}$ (i.e. the central matrices are exactly the scalar multiples of the identity). Please do this using $E_{ij}$'s.
\end{description}

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