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\author{Teschl Gerald}
\title{Quiz 3}
\date{11 Sep. 1995}

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\hfill {\bf Name}: $\ul{\hspace*{5cm}}$\\[2mm]
\hspace*{2mm}\hfill {\bf ID \#}: $\ul{\hspace*{5cm}}$\\[5mm]

{\Huge Quiz 3}\\[8mm]

\noindent 1) Which of the following points $(x,y)$ is a feasible solution to the system of inequalities $2x + 4y \ge 2$ and $x-y <0$?

a) (1,1),\hspace{3mm}b) (0,$\frac{1}{10}$),\hspace{3mm}c) ($\frac{1}{2}$,$\frac{1}{3}$),\hspace{3mm} d) (1,0),\hspace{3mm}e) (0,$\frac{1}{2}$)~\hfill (2pts)\\[2mm]

\noindent 2) Let
\bd
X = \left[ \ba{c} x_1 \\ x_2 \ea\right], \quad A = \left[ \ba{cc} a & b \\ c & d\ea\right], \quad D = \left[ \ba{c} e \\ f \ea\right], \quad I =\left[ \ba{cc} 1 & 0 \\ 0 & 1\ea\right],
\ed
where $a,b,c,d,e,f$ are numbers such that the matrices $A$, $I-A$, and $I+A$ are invertible. Solve the matrix equation
\bd
X = A X + D
\ed
for $X$.

a) $X = D (I-A)^{-1}$ b) $X=A^{-1}D$ c) $X=(I-A)^{-1}D$ d) $X= D(I+A)^{-1}$\\ e) $X=D A^{-1}$ \hfill (3pts)\\[8mm]

\noindent 3)
Suppose you have a {\em Leontief production model} consisting of an electric
company and a steel company. To produce a certain amount of electricity the electric
company needs 10\% of its own electricity and 5\% of the value of its own electricity
worth of steel. The steel company uses 15\% of its own steel, and 45\% of the value of
its own steel worth of electricity. Find the value of the produced electricity and the
value of the steel products if the customers demand 6 million dollars worth of electricity
and 4 million dollars worth of steel products.

\noindent 3.1) Write down the
\begin{itemize}
\item {\em output matrix} $X$, the {\em input-output matrix} $A$, the {\em consumer demand matrix} $D$,
\item and the {\em internal demand} matrix.~\hfill (3pts)
\end{itemize}
3.2) Compute the solution.~\hfill (2pts)

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