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\noindent
{\Large Math 211 \qquad Homework 1} {\hfill \bf Name: Sample Texfile}

\medskip\noindent
{\bf 1} (5 points) Find the point of intersection of the lines:
$x_1 + 2x_2 = -13, 3x_1 - 2x_2 = 1$.

Solution. To find the point of intersection, we solve the 
system with augmented matrix
$$\left[\matrix{
1 & 2 & -13 \cr
3 & -2  & 1 
}\right].$$
By row reduction, we get
$$\left[\matrix{
1 & 2 & -13 \cr
3 & -2  & 1 
}\right] \rightarrow \left[\matrix{
? & ? & ? \cr
? & ?  & ? 
}\right] .$$
Then ...
\vskip .3in


\medskip\noindent
{\bf 2} (5 points)
Solve the linear system:
\begin{eqnarray*} 
x_1 - 5x_2 + 4x_3 &=& -3 \cr
2x_1 - 7 x_2 + 3x_3 &=& -2 \cr
-2x_1+x_2+7x_3 & = & -1 \cr
\end{eqnarray*}

Solution. Consider augmented matrix
$$\left[\matrix{
1 & -5 & 4 & -3 \cr
2 & -7  & 3 & -2 \cr 
-2 & 1 & 7 & -1}\right].$$
By row reduction, we get .....

\vskip .3in

\medskip\noindent
{\bf 3} (5 points)
Solve the linear
system with augmented matrix
$$\left[\matrix{
2&0&-6&-8\cr
0&1& ~ 2& ~ 3\cr
3 & 6 & -2 & -4\cr}\right].$$

Solution. 
By row reduction, we get .....

\medskip\noindent
{\bf 4} (5 points) Find all the values $h$ so that the linear system with the following
augmented matrix is consistent:Applying row reduction to:
$$\left[\matrix{
1&h&-5\cr
2&-8&6\cr}\right].$$

Solution. By row reduction, we get ...

So, the system is consistent exactly when $h$ satisfies ...

\medskip\noindent
{\bf 5} (5 points) Row reduce the following augmented matrix. 
Determine the leading ones and pivot columns.
$$\left[\matrix{
1&2&4&5\cr
2&4&5&4\cr
4&5&4&3\cr}\right].$$

Soluion.

\vskip .5in
\medskip\noindent
{\bf 6} (5 points) Find the solutions of the system with augmented matrix:
$$\left[\matrix{
1&-2&-1&4\cr
-2&4&-5&6\cr}\right].$$

\vskip .5in
\medskip\noindent
{\bf 7} (5 points)  Find the solutions of the system with augmented matrix:
$$\left[\matrix{
1&0&-9&0&4\cr
0&1&3&0&-1\cr
0&0&0&1&-7\cr
0&0&0&0&0\cr}\right].$$

\vskip .5in
\medskip\noindent
{\bf 8} (5 points) Determine the conditions on $(h,k)$ so that the following system
has no solution, one solution, and infinitely many solutions.
$$x_1-3x_2 = 1, \quad 2x_1 + hx_2 = k.$$

Solution. Row reduce the augmented matrix...

\end{document}

