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\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
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\begin{document}


{\bf Question}

\begin{description}
\item[(a)] Solve the following system of equations
\begin{eqnarray*} x + 2y + 2z & = & 11 \\ 2x - y + z & = & 3 \\
-4x + 7y + z & = & 13 \end{eqnarray*} Give a geometrical
interpretation.
\item[(b)] Write down a 3x3 matrix which represents a
transformation in 3-dimensional space consisting of a rotation of
60$^\circ$ about the z-axis together with a magnification in the
z-direction by a scale factor of 2.  Write down its inverse and
check your answer by multiplication.
\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)] $\left.\begin{array}{ccc}1 & 2 & 2 \\ 2 & -1 & 1 \\ -4
& 7 & 1 \end{array} \right | \begin{array}{c} 11 \\ 3 \\ 13
\end{array} \rightarrow \left. \begin{array}{ccc} 1 & 2 & 2 \\ 0 &
-5 & -3 \\ 0 & 15 & 9 \end{array} \right| \begin{array}{c}
11\\-19\\57\end{array}$

$\begin{array}{rcl} {\rm So \ \ } x + 2y + 2z & = & 11 \\ 5y + 3z
& = & 19 \end{array}$

Let $z=t$ then $\ds y = \frac{19-3t}{5}$

Thus $\ds x = 11 - 2t - 2\left(\frac{19 -3t}{5} \right) = \frac{55
- 10t -38 +6t}{5} = \frac{17-4t}{5}$

So $\left( \begin{array}{c} x \\ y \\ z \end{array} \right) =
\left( \begin{array}{c} \frac{17}{5} \\ \frac{19}{5} \\ 0
\end{array} \right) + t \left( \begin{array}{c} \frac{-4}{5} \\
\frac{-3}{5} \\ 1 \end{array} \right) $

This system represents three plans meeting in a common line, whose
equation is the solution.

\item[(b)] $A = \left( \begin{array}{ccc} \frac{1}{2} &
\frac{-\sqrt3}{2} & 0 \\ \frac{\sqrt 3}{2} & \frac{1}{2} & 0 \\ 0
& 0 & 2 \end{array} \right)$

$A^{-1} = \left( \begin{array}{ccc} \frac{1}{2} & \frac{\sqrt3}{2}
& 0 \\ \frac{-\sqrt 3}{2} & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2}
\end{array} \right)$

Check $AA^{-1} = \left( \begin{array}{ccc} 1 &0 & 0 \\ 0 & 1 & 0
\\ 0 & 0 & 1 \end{array}\right)$
\end{description}


\end{document}
