Exam 5: Matrix Algebra and Applications

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { c c c c } 1 & 1 & - 1 & 1 \\- 3 & 1 & 1 & 0 \\- 3 & 1 & 0 & 1 \\8 & - 1 & - 1 & - 1\end{array} \right]$

Use the row reduction method to find the inverse, if it exists. ​ $A = \left[ \begin{array} { l l } 5 & 4 \\1 & 1\end{array} \right]$

What is the dimension of the matrix $A = \left[ \begin{array} { c c c } 2 & - 9 & 8 \\4 & 6 & - 7\end{array} \right]$ The dimension of the matrix is $\_\_\_\_\_\_\_\text { x }\_\_\_\_\_\_\_$ . What is the value of the element $b _ { 21 }$ $b _ { 21 } =\_\_\_\_\_\_\_$

Translate the given system of equations into matrix form. $\left\{ \begin{aligned}x - 5 y & = 9 + z \\- 8 x - 7 z & = 12 \\y & = - x - 3 z + 11\end{aligned} \right\}$

The table gives the number of people (in thousands) who visited Australia and South Africa in 1998: To Australia To South Africa From North America 440 190 From Europe 950 950 From Asia 1,790 200 Figures are rounded to the nearest 1,000. You predict that, in 2008, 20,000 fewer people from North America will visit Australia and 30,000 more will visit South Africa, 80,000 more people from Europe will visit each of Australia and South Africa, and 150,000 more people from Asia will visit South Africa, but there will be no change in the number visiting Australia. Use matrix algebra to predict the number of visitors from the three regions to Australia and South Africa in 2008.

The following matrix equation is equivalent to which system of linear equations $\left[ \begin{array} { c c c } 5 & - 1 & 4 \\0 & 2 & 3 \\1 & 5 & 0\end{array} \right] \cdot \left[ \begin{array} { l } x \\y \\z\end{array} \right] = \left[ \begin{array} { c } 50 \\- 30 \\90\end{array} \right]$

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 1 2 3 4 From 693.3 3.4 3,407.9 1,006.5 12.5 5.9 17.9 111.1 47.6 4.4 917.5 164.1 349.7 21 77.2 704.9 Total Output 9.471.5 676.3 6,755.5 4,653.8 How much additional production by the Sector 3 is necessary to accommodate a $100 increase in the demand for the products of Sector 1

If $A = \left[ \begin{array} { c c c } 2 & 0 & - 1 \\8 & 7 & 5\end{array} \right]$ , then find $3 A ^ { T }$ .

In the Second World War, during the struggle for New Guinea, intelligence reports revealed that the Japanese were planning to move a troop and supply convoy from the port of Rabaul at the Eastern tip of New Britain to Lae, which lies just west of New Britain on New Guinea. It could either travel via a northern route which was plagued by poor visibility, or by a southern route, where the visibility was clear. General Kenney, who was the commander of the Allied Air Forces in the area, had the choice of concentrating reconnaissance aircraft on one route or the other, and bombing the Japanese convoy once it was sighted. Suppose that General Kenney had a third alternative: Splitting his reconnaissance aircraft between the two routes Kenney's staff drafted the following outcomes for his choices, where the payoffs are estimated days of bombing time: Japanese Commander's Strategies Kenney's Strategies NorthernRoute SouthernRoute Northern Route 2 2.5 Split Reconnaissance 0.5 2 Southern Route 1 4 What would you have recommended to General Kenney __________ (Northern Route or Split Reconnaissance or Southern Route) What would you have recommended to the Japanese Commander __________ (Northern Route or Southern Route) How much bombing time results if these recommendations are followed __________days

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. $\left[ \begin{array} { c c c c } 1 & 1 & - 1 & 1 \\- 2 & 1 & 1 & 0 \\- 2 & 1 & 0 & 1 \\4 & - 1 & - 1 & - 1\end{array} \right]$

In January, the Left Coast Bookstore chain sold 600 hardcover books, 1,100 softcover books, and 1,800 plastic books in San Francisco; it sold 300 hardcover, 100 softcover, and 300 plastic books in Los Angeles. Hardcover books sell for $32 each, softcover books sell for $7 each, and plastic books sell for $11 each. Hard Soft Plastic San Francisco 600 1,100 1,800 Los Angeles 300 100 300 Use matrix multiplication to compute the total revenue at the two stores.

Your T-shirt operation is doing a booming trade. Last week you sold 70 tie-dyed shirts for $18 each, 55 Suburban State University crew shirts for $9 each, and 50 lacrosse T-shirts for $14 each. ​ Use matrix operations to calculate your total revenue for the week. ​ $R = \$$ __________

Use matrix inversion to solve the given system of linear equations. ​ 4x-7y-z =3 -z =1 x-2y+3z =-5 ​ $x=\frac{\text { }}{\text { }}$ ​ $y=\frac{\text { }}{\text { }}$ ​ $z=\frac{\text { }}{\text { }}$

Two sectors of some economy are Sector 1 and Sector 2. The input-output table involving these two sectors results in the following value for $( I - A ) ^ { - 1 }$ $( I - A ) ^ { - 1 } = \left[ \begin{array} { l l } 1.313 & 0.162 \\0.007 & 1.1331\end{array} \right]$ How many additional dollars worth of production of Sector 2 must be produced to meet a $1 increase in the demand for products of Sector 1 Round your answer to three decimal places.

Let $A = \left[ \begin{array} { c c c } 1 & \frac { 1 } { 3 } & 0 \\- 1 & 4 & - \frac { 1 } { 2 }\end{array} \right]$ and $B = \left[ \begin{array} { c c c } 2 & \frac { 1 } { 2 } & - 1 \\5 & - 6 & \frac { 2 } { 3 }\end{array} \right]$ . Evaluate $A + B$ .

Use the row reduction method to find the inverse of the matrix, if it exists. $A = \left[ \begin{array} { c c c } 1 & 3 & 0 \\- 1 & 3 & 0 \\3 & 0 & 2\end{array} \right]$

City Community College (CCC) plans to host Midtown Military Academy (MMA) for a wrestling tournament. Each school has three wrestlers in the 190 lb. weight class: CCC has Pablo, Sal, and Edison, while MMA has Carlos, Marcus and Noto. Pablo can beat Carlos and Marcus, Marcus can beat Sal and Edison, Sal can beat Carlos, Noto can beat Edison, while the other combinations will result in an even match. Set up a payoff matrix, and use reduction by dominance to decide which wrestler each team should choose as their champion. Does one school have an advantage over the other ?

Given the following matrix. $B = \left[ \begin{array} { c } \frac { 2 } { 3 } \\4 \\0 \\- \frac { 6 } { 7 }\end{array} \right]$ What are the dimensions $\_\_\_\_\_\_\_\text { x }\_\_\_\_\_\_\_$ What is the value of the element $b _ { 31 }$ $b _ { 31 } =\_\_\_\_\_\_\_$

Given $A = \left[ \begin{array} { c c } 9 & 3 \\- 8 & 2\end{array} \right]$ , $B = \left[ \begin{array} { l l } 8 & 4\end{array} \right]$ , and $C = \left[ \begin{array} { c c c } - 1 & 5 & 7 \\8 & 0 & 4\end{array} \right]$ , which of the following can be calculated Can $( B A ) C$ be calculated __________ Can $B C + A$ be calculated __________ Can $B ( A C )$ be calculated __________ Can $B C - A$ be calculated __________

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 160 tie-dyed shirts for $10 each, 95 (low quality) crew shirts at $5 apiece, and 50 lacrosse T-shirts for $7 each. ​ Use matrix operations to calculate her total revenue for the week. ​ $R = \$$ __________