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Matrices are a convenient and compact way to organize information. For example, let’s say it’s Valentine’s day, and you are surprising your crush with flowers. You go to the store and have $40 in cash to spend. You know your crush’s favorite flowers are tulips and lilies. If tulips are $2 each and lilies are $1 each, which of the following matrix products correctly represents spending all of your money?
You are gathering data on how much you and your classmates spend on textbooks for school. You bought 1 English, 1 Math, and 1 Science textbook for a total cost of $215. Your friend bought 2 Math textbooks [one for themselves and one for their brother] and a Science textbook for a total cost of $173. Another friend of yours only bought a Science textbook for $80. Which of the following augmented matrices correctly represents this situation?
Use the following information to answer questions 3 and 4
Exchange rates between different currencies are often expressed as a matrix. The matrix A describing the exchange rate of US Dollars [$] and British Pounds [£] is the following 2×2:
\(\begin{array}{cc}
\$ & £ \\
\end{array}\)
\(\left[
\begin{array}{cc}
1 & 1.27 \\
0.79 & 1 \\
\end{array}
\right]
\begin{array}{c}
\$ \\
£ \\
\end{array}\)
What does the matrix product AB represent?
Photo by Sebastian Molina fotografía on Unsplash
You are a spy in the field and periodically have to update headquarters on your location, expressed as latitudinal \(\left(x_1\right) \) and longitudinal \(\left(x_2\right) \) coordinates:\( \left[
\begin{array}{c}
x_1 \\
x_2 \\
\end{array}
\right]\). However, you have to encode your message so that if your enemies intercept your message, they do not discover your actual location. You and headquarters agree on a transformation matrix A that will encode your location:
\(\left[
\begin{array}{cc}
3 & 1 \\
4 & – 2 \\
\end{array}
\right]\)
If your actual current location is at \( \left[
\begin{array}{c}
9 \\
7 \\
\end{array}
\right]\) (which puts you in Nigeria) what encoded coordinates should you send in your message? Write your answer in the format \( \left(y_1,y_2\right)\) where \(y_1\) is latitude and \(y_2\) is longitude. (Example: \( \left[
\begin{array}{c}
3 \\
5 \\
\end{array}
\right]\)=(3, 5))
[Master Question] Headquarters changes the transformation matrix once a week as a safety precaution. If this week’s encoding matrix is \(A=\left[
\begin{array}{cc}
– 1 & 6 \\
2 & 0 \\
\end{array}
\right]\), and they receive your encoded coordinates as (-272,-32). What country are you actually in?
Note: first, find your actual coordinates \(x_1\), and \(x_2\). Then, head to Google Maps here and type your answer coordinates in the following format in the search bar: \(x_1\),\(x_2\) [hint: you can turn the matrices into a system of equations and solve from there]. You might need to zoom out to see the country’s name.