Exam 5: Matrix Algebra and Applications

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. foot 780 820 300 Number of sq. ft 510 800 190 Use matrix multiplication to estimate the total cost of the real estate. $\cos t = \$$ __________ $780$ $780$

Determine which pair of matrices is an inverse pair.

Use matrix inversion to solve the given system of linear equations. 5x+y=1 19x+4y=3

Use row reduction to find the inverse of the given matrix, if it exists, and check your answer by multiplication. ​ $\left[ \begin{array} { c c c c } 1 & 1 & - 1 & 1 \\2 & 1 & 1 & 0 \\2 & 1 & 0 & 1 \\0 & - 1 & - 1 & 3\end{array} \right]$

Which of the following is a true statement

Given the technology matrix A, and an external demand vector D, find the production vector X. ​ ​ $A = \left[ \begin{array} { c c c } 0.25 & 0.25 & 0.25 \\0.25 & 0.6 & 0.25 \\0.25 & 0.25 & 0.25\end{array} \right]$ , $D = \left[ \begin{array} { c } 8,000 \\16,000 \\8,000\end{array} \right]$

Each unit of television news requires 0.5 units of television news and 0.4 units of radio news. Each unit of radio news requires 0.7 units of television news and no radio news. With sector 1 as television news and sector 2 as radio news, set up the technology matrix A.

Solve the system using the inverse matrix. x+3y+z =19 x+y+z =7 4x+y+z =25

Find the inverse of the given matrix, if it exists. ​ $\left[ \begin{array} { c c c } 1 & 1 & 1 \\1 & 0 & 9 \\1 & - 1 & 1\end{array} \right]$

Solve the system using the inverse matrix. x+2y+z =20 x+y+z =13 4x+y+z =19 $x=\frac{\text { }}{\text { }}$ $y=\frac{\text { }}{\text { }}$ $z=\frac{\text { }}{\text { }}$

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 $( I - A ) ^ { - 1 }$ $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c } 1.21 & 0.155 \\0.002 & 1.1242\end{array} \right]$ 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

You and your friend have come up with the following simple game to pass the time: at each round, you simultaneously call "heads" or "tails". If you have both called the same thing, your friend wins one point; if your calls differ, you win one point. Set up the payoff matrix.

Let $( I - A ) ^ { - 1 } = \left[ \begin{array} { c c c } 1.6 & 0.1 & 0 \\0.2 & 1.1 & 0.2 \\0.1 & 0.4 & 1.2\end{array} \right]$ and assume that the external demand for the products in Sector 1 increases by 1 unit. By how many units should each sector increase production

Editors' workloads were increasing during the 1990s, as the following table shows. Books/Editor 2.7 4 4.8 5.3 Editors 18,000 16,000 12,500 13,000 ​ Write a matrix expression which would estimate the total number of books edited during the years 1993-1996.

Compute the product of the two matrices (if possible). ​ $\left[ \begin{array} { l l l } - 1 & 4 & 2\end{array} \right] \cdot \left[ \begin{array} { c } 13 \\2 \\- 5\end{array} \right]$

Find the inverse of the given matrix, if it exists. $\left[ \begin{array} { c c c } 1 & 1 & 1 \\1 & 0 & 5 \\1 & - 1 & 1\end{array} \right]$

Is the following statement If D and B are matrices and $D B = 1$ , then $B D = 1$ .

Your Abercrom B men's fashion outlet has a 40% chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a 60% chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a 40% chance that Abercrom A will opt for the empty shirt option. The payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation. What is the expected resulting effect on your customer base Round the answer to the nearest whole number.

Find the matrix product, if possible. ​ $\left[ \begin{array} { c c } 1 & - 5 \\0 & 3 \\0 & - 2\end{array} \right] \cdot \left[ \begin{array} { c c c } 4 & 0 & - 1 \\5 & - 5 & 2\end{array} \right]$ ​ If the product is not defined please enter undefined.

Is the pair of matrices below the inverse pair $\left[ \begin{array} { l l } 0 & 4 \\4 & 0\end{array} \right]$ , $\left[ \begin{array} { c c } 0 & - \frac { 1 } { 4 } \\- \frac { 1 } { 4 } & 0\end{array} \right]$