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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Problem Set 3}
\centerline{{\bf due:} September 21, 2012}
\vskip 10pt
\noindent
{\it `A' Section}
\bigskip
\begin{enumerate}
\item[1.]  Let $\mathcal{A} = \{a,b,c\}$.  In class we discussed the 
set $\mathcal{S}(\mathcal{A})$ of permutations of $\mathcal{A}$, and we wrote 
\begin{equation}
\label{perm}
\mathcal{S}(\mathcal{A}) = \{I_{\mathcal{A}}, R_a,R_b,R_c,C_1,C_2\}
\end{equation}
where the $R$'s were mappings that fixed one element and swapped the other two, 
and the $C$'s were the cyclic permutations.  
For instance $R_a(a) = a$, $R_a(b) = c$, and $R_a(c) = b$ and 
$C_1(a) = b$, $C_1(b) = c$ and $C_1(c) = a$.  We know that $\circ$ defines
a binary operation on $\mathcal{S}(\mathcal{A})$ by results from class on 
September 14 and 17.  Make the operation table for $\circ$ on this set, showing
the results of composing all pairs of elements of $\mathcal{S}(\mathcal{A})$, listing
the labels for the rows and columns in the order given in Eq.~\eqref{perm} above.
The entry in the row with label $f$ and the column with label $g$ should be 
the mapping $f \circ g$ in all cases. 
\item[2.] Let 
$$A =  \begin{pmatrix} 1 & 1 & -1\\
                      -2 & 3 & 0\end{pmatrix},\  
B = \begin{pmatrix} 1 \\ 7 \\ 3 \end{pmatrix},\ 
C = \begin{pmatrix} 3 & -3 & -4\\
                    1 & -8 & 2\\
                    0 & 1 & 4 \end{pmatrix},\  
D = \begin{pmatrix} 4 & 3 & 3\\
                    5 & 0 & -2\end{pmatrix}.
$$
Perform the indicated matrix operations, if possible.  If the operation
is not possible, say why not.
\begin{enumerate}
\item[(a)]  $A + D$
\item[(b)]  $AC$
\item[(c)]  $AD + A$
\item[(d)]  $CB + AB$
\end{enumerate}
\item[3.]  Let $S$ be the following set of matrices in $M_{3\times 3}(\mathbb{R})$:
\begin{equation}
\label{mats}
S = \{I_3,F_x,F_y,F_z,G_1,G_2\},
\end{equation}
where $I_3$ is the $3\times 3$ identity matrix, 
$$F_x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix},\  
F_y = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0\\ 1 & 0 & 0 \end{pmatrix},\  
F_z = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix},\  
$$
and 
$$G_1 = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix},\  
G_2 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}.$$
Show that \emph{matrix multiplication} defines a binary operation on this set $S$
by constructing an operation table showing the results of doing all possible
products of pairs of elements in $S$.  Label the rows and columns in the order
given in Eq.~\eqref{mats} above.  Do you notice something about this table
and the one in question 1?
\end{enumerate}
\vskip 10pt
\noindent
{\it `B' Section}
\vskip 10pt
\begin{enumerate}
\item[1.]  Let $\mathcal{A}$ be a set and $f,g \in S(\mathcal{A})$ be 
permutations.  
\begin{enumerate}
\item[(a)]  Show that $(f^{-1})^{-1} = f$ as mappings on $\mathcal{A}$.
\item[(b)]  Show that $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$ as mappings on $\mathcal{A}$.  
\end{enumerate}
\item[2.]  Suppose that $\mathcal{A}$ is a \emph{finite} set.  Show:
\begin{enumerate}
\item[(a)]  A mapping $f : \mathcal{A} \to \mathcal{A}$ is a permutation
of $\mathcal{A}$ if and only if $f$ is one-to-one.
\item[(b)]  A mapping $f : \mathcal{A} \to \mathcal{A}$ is a permutation
of $\mathcal{A}$ if and only if $f$ is onto.
\item[] Hint for both parts:  The idea here is to show that each of the 
defining properties of a permutation \emph{implies the other} in 
this special situation.  The ideas used in the proof of Exercise B 2 on 
Problem Set 1 may be helpful.  
\end{enumerate}
\item[3.]  Show that matrix addition and matrix multiplication both
define binary operations on the set of matrices 
$$M = \left\{\begin{pmatrix} a & -b \\ b & a\end{pmatrix} \mid a,b \in \mathbb{R}\right\}.$$
(That is, show that if $A, B$ are two general matrices in $M$, 
then $A + B \in M$ and $A\cdot B \in M$.)  
\item[4.]  Let 
$$A = \begin{pmatrix} a & b \\ c & d\end{pmatrix}$$ 
be any matrix
in $M_{2\times 2}(\mathbb{R})$ such that $\Delta = ad - bc \ne 0$.  What happens
if you multiply $AB$ and $BA$ for the matrix
$$B = \begin{pmatrix} \frac{d}{\Delta} & -\frac{b}{\Delta} \\ -\frac{c}{\Delta} & \frac{a}{\Delta}
\end{pmatrix}?$$
What does this say about $B$?  
\end{enumerate}


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