Reduce the given payoff matrix by dominance. $$ \left.\begin{array}{rrr} p & q & r \\ a & 2 & 0 & 10 \\ b & 15 & -4 & -5 \end{array}\right] $$

Can an external demand be met by an economy whose technology matrix \(A\) is the identity matrix? Explain.

Foreclosure Crisis Starting in \(2010,\) on the heels of the \(2007-\) 2009 subprime mortgage crisis, the United States saw an epidemic of mortgage foreclosures, often initiated improperly by large financial institutions. Based on the following table, which shows the numbers of foreclosures in three states during three months of \(2011:^{1}\) $$ \begin{array}{|r|c|c|c|} \hline & \text { June } & \text { July } & \text { Aug. } \\ \hline \text { California } & 54,100 & 56,200 & 59,400 \\ \hline \text { Florida } & 23,800 & 22,400 & 23,600 \\ \hline \text { Texas } & 9,300 & 10,600 & 10,100 \\ \hline \end{array} $$ Each month, your law firm handled \(10 \%\) of all foreclosures in California, \(5 \%\) of all foreclosures in Florida, and \(20 \%\) of all foreclosures in Texas. Use matrix multiplication to compute the total number of foreclosures handled by your firm in each of the months shown.

Rotations If a point \((x, y)\) in the plane is rotated counterclockwise about the origin through an angle of \(60^{\circ},\) its new coordinates \(\left(x^{\prime}, y^{\prime}\right)\) are given by $$ \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=S\left[\begin{array}{l} x \\ y \end{array}\right] $$ where \(S\) is the \(2 \times 2\) matrix $\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\( and \)a=1 / 2$ and \(b=\sqrt{3 / 4} \approx 0.8660 .\) a. If the point (2,3) is rotated counterclockwise through an angle of \(60^{\circ},\) what are its (approximate) new coordinates? b. Referring to Exercise \(61,\) multiplication by what matrix would result in a counterclockwise rotation of \(105^{\circ} ?\) (Express the matrices in terms of \(S\) and the matrix \(R\) from Exercise 61.) [HINT: Think of a rotation through \(105^{\circ}\) as a rotation through \(60^{\circ}\) followed by a rotation through \(\left.45^{\circ} .\right]\) c. Multiplication by what matrix would result in a clockwise rotation of \(60^{\circ}\) ?

If you think of numbers as \(1 \times 1\) matrices, which numbers are invertible \(1 \times 1\) matrices?

Mexico Economy Economists generally divide a country's economy into three broad sectors: primary, secondary, and tertiary. The primary sector is the sector using natural resources to produce raw materials, and includes oil extraction, agriculture, mining, and fishing. The secondary or industrial sector is the sector that uses raw materials to produce manufactured goods, and includes oil refining, textiles, and electronics. The tertiary or services sector provides services, and includes tourism, financial services, and health care. Exercises 25-30 are based on the following technology matrix for Mexico in \(2008 .\) (The sectors are in the order primary, secondary, and tertiary, and entries are rounded to two decimal places.) \(^{36}\) $$A=\left[\begin{array}{lll}0.09 & 0.03 & 0.00 \\\0.14 & 0.23 & 0.08 \\\0.07 & 0.12 & 0.15\end{array}\right]$$ Determine how the three sectors of the Mexico economy would react to an increase in demand for tourism (tertiary sector) of 1,000 billion pesos and a decrease in the other two sectors of 1,000 billion pesos each.