Exam 5: Matrix Algebra and Applications

Karen Sandberg, your competition in Suburban State U's T-shirt market, has apparently been undercutting your prices and outperforming you in sales. Last week she sold 120 tie-dyed shirts for $12 each, 80 (low quality) crew shirts at $1 apiece, and 50 lacrosse T-shirts for $6 each. Use matrix operations to calculate her total revenue for the week. ​

In January, the Left Coast Bookstore chain sold 900 hardcover books, 1,200 softcover books, and 1,700 plastic books in San Francisco; it sold 600 hardcover, 500 softcover, and 700 plastic books in Los Angeles. Hardcover books sell for $33 each, softcover books sell for $9 each, and plastic books sell for $13 each. Hard Soft Plastic San Francisco 900 1,200 1,700 Los Angeles 600 500 700 ? Use matrix multiplication to compute the the total revenue at the two stores. Please enter your answer as a $2 \times 1$ column matrix in the following form: $\left[ \begin{array} { l l l l } \text { San Francisco Profit}\\ \text {Los Angeles Profit }\\\end{array} \right]$

Microbucks Computers makes two computers, the Pomegranate II and the Pomegranate Classic, at two different factories. The Pom II requires 2 processor chips, 18 memory chips, and 40 vacuum tubes, and the Pom Classic requires 1 processor chip, 2 memory chips, and 30 vacuum tubes. At the beginning of the year, Microbucks has in stock 600 processor chips, 9,000 memory chips, and 14,000 vacuum tubes at the Pom II factory and 500 processor chips, 2,000 memory chips, and 40,000 vacuum tubes at the Pom Classic factory. It manufactures 50 Pom IIs and 50 Pom Classics each month. ​ Find the company's inventory of parts after 2 months, using matrix operations. ​ Please enter your answer as a matrix in the following form (Do not use commas.): ​ $\left[ \begin{array} { c c c c } \text { Pom II processors Pom II memorychips Pom II vacuum tubes } \\\text { Pom } C \text { processors Pom } C \text { memorychips Pom } C \text { vacuum tubes }\end{array} \right]$

Let $A = \left[ \begin{array} { c c c } 0 & \frac { 1 } { 2 } & - \frac { 2 } { 3 } \\- 1 & 2 & \frac { 1 } { 3 }\end{array} \right]$ and $B = \left[ \begin{array} { c c c } - 2 & 6 & 0 \\2 & \frac { 1 } { 2 } & - 1\end{array} \right]$ . Evaluate $3 A - 4 B$ .

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. ​ $\left[ \begin{array} { c c c c } 1 & 1 & - 1 & 1 \\- 1 & 1 & 1 & 0 \\- 1 & 1 & 0 & 1 \\4 & - 1 & - 1 & - 1\end{array} \right]$

Compute the product of the two matrices (if possible). $\left[ \begin{array} { l l l } 4 & 5 & - 1\end{array} \right] \cdot \left[ \begin{array} { c } - 2 \\3 \\8\end{array} \right]$

A manufacturer of electrical machinery is located in a cramped, though low-rent, factory close to the center of a large city. The firm needs to expand, and it could do so in one of three ways: (1) remain where it is and install new equipment, (2) move to a suburban site in the same city, or (3) relocate in a different part of the country where labor is cheaper. Its decision will be influenced by the fact that one of the following will happen: (I) the government may introduce a program of equipment grants, (II) a new suburban highway may be built, or (III) the government may institute a policy of financial help to companies who move into regions of high unemployment. The value to the company of each combination is given in the following payoff matrix. I II III 200 150 140 130 220 130 110 110 220 If the manufacturer judges that there is a 60% probability that the government will go with option I, a 30% probability that they will go with option II, and a 10% probability that they will go with option III, what is the manufacturer's best option

Solve for x, y, and z. ​ x-y 3 z x+y = -1 3 4 7 x=\_\_\_\_\_\_\_ y=\_\_\_\_\_\_\_ z=\_\_\_\_\_\_\_

The Left Coast Bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of books: hardcover, softcover, and plastic (for infants). The table shows the number of books in stock at the beginning of January. Hard Soft Plastic San Francisco 1,000 2,000 5,000 Los Angeles 1,000 5,000 2,000 Suppose its sales in January were as follows: 700 hardcover books, 1,300 softcover books, and 2,000 plastic books sold in San Francisco, and 400 hardcover, 300 softcover, and 500 plastic books sold in Los Angeles. Now suppose that the stores maintained the same sales figures for the first 6 months of the year. Each month the chain restocked the stores from its warehouse by shipping 900 hardcover, 1,600 softcover, and 1,800 plastic books to San Francisco and 700 hardcover, 700 softcover, and 200 plastic books to Los Angeles. Use matrix operations to determine the inventory in each store at the end of June.

The table shows the cost of one square foot of residential real estate, in dollars per square foot, together with the number of square feet your development company intends to purchase in each city. New York London Hong Kong Cost per sq. ft 780 810 270 Number of sq. ft 500 780 740 Use matrix multiplication to estimate the total cost of the real estate. 780 780 780 780

The total amount of cheese, in billions of pounds, produced in the western and north central states in 1999 and 2000 was as follows. Western States 3.0 3.0 North Central States 3.9 4.1 Thinking of this table as a (labeled) $2 \times 2$ matrix P, compute the matrix product $\left[ \begin{array} { l l } - 1 & 1\end{array} \right] \cdot P$ .

Your T-shirt operation is doing a booming trade. Last week you sold 60 tie-dyed shirts for $17 each, 75 Suburban State University crew shirts for $8 each, and 30 lacrosse T-shirts for $11 each. Use matrix operations to calculate your total revenue for the week. ​

Decide whether the game is strictly determined. B pqr a b -2 2 -5 -2 4 -2 __________ (answer strictly determined or not strictly determined) If it is, what are the players' optimal pure strategies A __________ B __________ What is the value of the game __________

Given the technology matrix A, and an external demand vector D, find the production vector X. ​ $A = \left[ \begin{array} { c c } 0.2 & 0.3 \\0 & 0.4\end{array} \right]$ , $D = \left[ \begin{array} { l } 11,000 \\15,000\end{array} \right]$

Four sectors of some economy are (1) Sector 1, (2) Sector 2, (3) Sector 3, and (4) Sector 4. The input-output table involving these four sectors was as follows (all figures are in millions of dollars) To from 11,534 8 100 868 23,307 4,159 0 4,931 0 0 403 66 5,281 10,934 3,285 223 Total Output 99,840 116,189 13,786 45,799 Determine how these four sectors would react to an increase in demand for Sector 1 production of $1,000 million.

Editors' workloads were increasing during the 1990s, as the following table shows. Books/Editor 2.9 4 4.8 5.5 Editors 20,000 18,000 9,500 10,000 Which matrix expression would estimate the total number of books edited during the years 1993-1996

Given the technology matrix A, and an external demand vector D, find the production vector X. $A = \left[ \begin{array} { l l } 0.4 & 0.3 \\0.1 & 0.2\end{array} \right]$ , $D = \left[ \begin{array} { l } 30,000 \\25,000\end{array} \right]$

Given the following matrix. $A = \left[ \begin{array} { l l l l } - 2 & 4 & 0 & 3\end{array} \right]$ What are the dimensions __________ $x$ __________ What is the value of the element $Q _ { 14 }$ $a _ { 14 } =$ __________

Let $A = \left[ \begin{array} { c c c c } 5 & - 5 & 4 & - 4 \\5 & 1 & 0 & 0\end{array} \right]$ and $B = \left[ \begin{array} { c c c c } 2 & 5 & - 1 & - 3 \\2 & 2 & 0 & - 2\end{array} \right]$ . Evaluate A + B.

Use row reduction to find the inverse of the given matrix, if it exists. Check your answer by multiplication. $\left[ \begin{array} { c c c } 1 & 9 & 10 \\0 & 9 & 10 \\1 & 0 & 1\end{array} \right]$