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\centerline{Mathematics 243--Algebraic Structures, section 1}
\centerline{Discussion 2 -- Matrices, and Matrix Sums and Products}
\centerline{September 19, 2003}
\bigskip
\noindent
{\it Background}
\bigskip
Today, we will introduce a new class of mathematical objects
called {\it matrices} and sum and product operations on them.
This will give us some important additional examples of binary
operations to look at.  (And we will also use matrices extensively
next semester in Linear Algebra.) In general, a matrix is a {\it rectangular
array of numbers}.  There can be any number of rows (horizontal)
and columns (vertical) in a matrix.  But to begin with simple
cases, we will only consider matrices with at most 2 rows or columns.
For instance in the following equations
$$A = \pmatrix{1&-8\cr
               7& 6\cr}, \qquad  B = \pmatrix{12&-1\cr}, \qquad 
               C=\pmatrix{1\cr
                          -2\cr},$$
$A$ has 2 rows and 2 columns (we also say $A$ is a $2\times 2$ matrix),
$B$ has 1 row and 2 columns, and $C$ has 2 rows and 1 column.

Let $M_{mn}$ be the set of all $m\times n$ matrices (with real entries).
We can define a binary sum operation on $M_{mn}$ by adding the
corresponding entries in the matrices.  For instance with $m = n = 2$:
$$\pmatrix{1&-8\cr
          7& 6\cr} 
+ \pmatrix{0&2\cr
           1& 2\cr} 
= \pmatrix{1+0 & -8+2\cr
           7+1 &  6+2\cr}
= \pmatrix{1 & -6\cr
           8 & 8\cr}$$
On the other hand, if two matrices have different sizes they just
{\it can't be added} -- the sum is not defined in that case.

Two matrices $A,B$ can be multiplied to produce
another matrix $C = AB$ if the number of columns of $A$ is the same
as the number of rows of $B$.  The product then has the same 
number of rows as $A$ and the same number of columns as $B$.
The entries of the product matrix can be formed by taking {\it dot
products} of the rows of $A$ with the columns of $B$ as in 
the following example:
$$\eqalign{
\pmatrix{1&-8\cr
         7& 6\cr} 
\cdot \pmatrix{0&2\cr
              1& 2\cr} 
&= 
\pmatrix{1\cdot 0 + -8\cdot 1 & 1\cdot 2 + -8\cdot 2\cr
         7\cdot 0 + 6\cdot 1 & 7\cdot 2 + 6 \cdot 2\cr}\cr
&=\pmatrix{-8 & -14\cr
            6 & 26\cr}
\cr} \leqno(1)$$
This product defines a binary operation on the set $M_{22}$ of
all $2\times 2$ matrices.

\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\item{A)}  Show that the sum operation on $M_{12}$ or $M_{21}$
is essentially the same as the vector sum operation on 
${\bf R}^2$ you have studied in MATH 241.
\bigskip
\item{B)} 
\itemitem{1)} Is the matrix sum operation on $M_{22}$ commutative?
Prove or give a counterexample.
\itemitem{2)}  Is it associative?  Prove or give a counterexample.
\itemitem{3)}  Is there an identity element?  If so, what matrix
is it?  If not, say why not.
\itemitem{4)}  Does every element have an inverse under this operation? 
Prove or give a counterexample. 
\bigskip
\item{C)}
\itemitem{1)}  Compute the matrix product                 
$$\pmatrix{0&2\cr 1& 2\cr} \cdot
\pmatrix{1&-8\cr 7& 6\cr}$$
(same matrices as in (1) above, but in the opposite order).
What does this say about the product operation on $M_{22}$?
\itemitem{2)}  Is the product on $M_{22}$ associative?  Prove
or give a counterexample.
\itemitem{3)}  Show that the matrix 
$$I = \pmatrix{1 & 0 \cr 0 & 1\cr}$$
is an identity element for matrix multiplication in $M_{22}$.
\itemitem{4)}  Does the matrix $A = \pmatrix{0&2\cr 1& 2\cr}$
have a multiplicative inverse?  Try to find one by 
solving for the entries in $B = \pmatrix{a&b\cr c& d\cr}$
to make $AB = I = BA$.
\bigskip
\noindent
{\it Assignment}
\bigskip
Group write-ups due in class on Wednesday, September 24.
\bye
                     
          
                       
