Using Matrices to Represent, Manipulate Data - Precalculus

A clothing store lists prices (in dollars) for two items (T-shirt, Hoodie) at three locations (North, Central, South). Rows represent items and columns represent locations. The current price matrix is $P=\begin{bmatrix}20&25&30\\30&35&40\end{bmatrix}.$ All prices increase by 10%. Based on the matrix, what is the entry in row 1, column 3 of the new price matrix $1.1P$ ?
Answer: $33 This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B$ = total from both sources), subtraction finds changes or differences ( $A - B$ = net change), scalar multiplication scales all values uniformly ( $kA$ = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar 1.1 means increasing all prices by 10%, so the new matrix is 1.1P where each entry is multiplied: row 1, column 3 gives $1.1 \times 30 = 33$ , giving the new price of T-shirts at the South location. Choice A is correct because it accurately calculates and interprets the result of the scalar multiplication for the specified entry. Choice B suggests using the original price without the increase, stating $30 when the context requires multiplying by 1.1 to get $33. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities.
A city transit system uses an adjacency matrix $M$ to represent direct train connections among 3 stations (rows = starting station, columns = destination station). Entry $(i,j)$ is 1 if there is a direct train from station $i$ to station $j$ , and 0 otherwise: $M=\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}$ For the network described, what does the matrix $M^2$ represent?
Answer: The number of 2-train routes (two-step connections) from each station to each station This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrices provide a compact way to organize related data: rows and columns represent different categories (like products and stores, or students and test scores), and each entry (i, j) represents a specific data point (like the inventory of product i at store j). The product $M^2$ combines the relationships: rows of first M (starting stations) and columns of second M (destinations) relate, and entry (i, j) of $M^2$ represents number of two-step paths from i to j, calculated as row-column dot product example for (1,1): $[0,1,0] \cdot [0,0,1] = 0 \times 0 + 1 \times 0 + 0 \times 1 = 0$ . Choice A is correct because it correctly interprets the entry in context. Choice B interprets the matrix operation incorrectly, claiming direct connections when $M^2$ actually represents two-step connections. For matrix multiplication in data contexts, think about how dimensions relate: if A is $products \times stores$ (showing inventory) and B is $stores \times warehouses$ (showing shipments), then AB is $products \times warehouses$ (showing total by product and destination). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by $1.2$ for 20% increase), and matrix multiplication for combining unit costs with quantities.
A gym records the number of class sign-ups for two class types across three days. Rows represent class type (Yoga, Spin) and columns represent days (Mon, Wed, Fri). Matrix $M$ shows sign-ups (in people): $M=\begin{bmatrix}25&20&30\\15&10&20\end{bmatrix}.$ Using the matrix, what is the entry in row 2, column 3 of $M$ , and what does it represent?
Answer: $20$ people; Spin sign-ups on Friday This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrices provide a compact way to organize related data: rows and columns represent different categories (like products and stores, or students and test scores), and each entry $(i, j)$ represents a specific data point (like the inventory of product $i$ at store $j$ ). Entry $(2, 3)$ of matrix $M$ represents Spin sign-ups on Friday, so entry $(2, 3) = 20$ means $20$ people signed up for Spin class on Friday. Choice A is correct because it accurately calculates and interprets the result with the correct value, class type, and day. Choice B makes an arithmetic error, using the value from row 1 column 3 ( $30$ ) instead of row 2 column 3 ( $20$ ). Key to matrix data problems: first understand what each row and column represents, then determine what each entry $(i, j)$ means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is $25$ ,' say 'the remaining inventory is $25$ units' or 'the new price is $25$ ' to show you understand what the numbers represent.
A coffee shop tracks the number of pastries baked (in units) each morning at two locations. Rows represent pastry type (Row 1 = muffins, Row 2 = croissants, Row 3 = scones) and columns represent location (Column 1 = Downtown, Column 2 = Uptown). Morning bake counts are $A=\begin{bmatrix}20 & 30\\15 & 25\\10 & 35\end{bmatrix}$ Units sold by noon are $B=\begin{bmatrix}5 & 10\\10 & 15\\5 & 20\end{bmatrix}$ Based on the matrices, what does the matrix $A-B$ represent in this context?
Answer: The remaining pastries (in units) after noon sales for each pastry type at each location This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources (A + B = total from both sources), subtraction finds changes or differences (A - B = net change), scalar multiplication scales all values uniformly (kA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The matrix A - B represents the remaining inventory after sales, calculated entry-by-entry: for muffins at Downtown, we have 20 - 5 = 15 units remaining, meaning 15 muffins are left unsold after the noon sales. Choice B is correct because it properly identifies that A - B gives the remaining pastries after subtracting the units sold from the units baked. Choice D suggests using a percentage calculation when the context requires simple subtraction: to find remaining inventory, we need to subtract sold from baked, not calculate percentages. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is 15,' say 'the remaining inventory is 15 units' to show you understand what the numbers represent.
If the company wants to create a matrix showing the number of cities that can be reached directly from each city, which operation should be performed?
Answer: Calculate $R \cdot \mathbf{1}$ where $\mathbf{1}$ is a column vector of ones To count reachable cities from each city, we sum each row of matrix $R$ . This is accomplished by multiplying $R$ by a column vector of ones, giving $R \cdot \mathbf{1}$ , which results in a column vector where each entry is the sum of the corresponding row. Choice B incorrectly references the trace (sum of diagonal elements). Choice C gives the total number of outgoing routes from all cities combined. Choice D gives back the original matrix $R$ since multiplying by ones element-wise doesn't change values.
A café lists prices (in dollars) for two drinks at three locations. Rows represent drinks (Coffee, Tea) and columns represent locations (Downtown, Mall, Campus). The current price matrix is $P=\begin{bmatrix}5&5&10\\5&10&10\end{bmatrix}.$ All prices increase by 10%. Based on the matrix, which calculation gives the new price matrix?
Answer: $1.10P$ This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B = \text{total from both sources}$ ), subtraction finds changes or differences ( $A - B = \text{net change}$ ), scalar multiplication scales all values uniformly ( $kA = \text{all values scaled by } k$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar $1.10$ means increasing all prices by 10%, so the new matrix is $1.10P$ where each entry is multiplied: for example, row 1 column 1 becomes $1.10 \times 5 = 5.50$ dollars. Choice B is correct because it properly identifies the matrix operation needed and accurately calculates the uniform scaling for a percentage increase. Choice A applies the percentage incorrectly, adding 0.10 instead of multiplying by $1 + 0.10$ as decimal. Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by $1.2$ for 20% increase), and matrix multiplication for combining unit costs with quantities. When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
A transit planner uses an adjacency matrix $M$ for a network of three stations (1, 2, 3). Entry $(i,j)$ is $1$ if there is a direct route from station $i$ to station $j$ , and $0$ otherwise: $M=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$ For the network described, what does the matrix $M^2$ represent in this context?
Answer: The number of 2-step routes from station $i$ to station $j$ in entry $(i,j)$ This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources (A + B = total from both sources), subtraction finds changes or differences (A - B = net change), scalar multiplication scales all values uniformly (kA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The product M² combines the relationships: rows and columns of M represent stations, and entry (i, j) of M² represents the number of 2-step routes from station i to station j, calculated as row-column dot product example for (1,2): row1 dot column2 = $0 \times 0 + 1 \times 0 + 0 \times 1 = 0$ . Choice B is correct because it correctly interprets the entry in context for the squared adjacency matrix. Choice A interprets the matrix operation incorrectly, claiming it shows direct routes when M² actually represents 2-step routes. For matrix multiplication in data contexts, think about how dimensions relate: if A is products×stores (showing inventory) and B is stores×warehouses (showing shipments), then AB is products×warehouses (showing total by product and destination). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities.
A café tracks cups of coffee sold (in units) for two drink types (Hot, Iced) across three days (Mon, Tue, Wed). Matrix $A$ is cups sold in Week 1 and matrix $B$ is cups sold in Week 2: $A=\begin{bmatrix}20&25&30\\15&20&25\end{bmatrix},\quad B=\begin{bmatrix}10&15&20\\5&10&15\end{bmatrix}.$ Based on the matrices, what does entry $(2,3)$ of $A+B$ represent?
Answer: The total cups (in units) of Iced coffee sold on Wednesday across Weeks 1 and 2 This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B = \text{total from both sources}$ ), subtraction finds changes or differences ( $A - B = \text{net change}$ ), scalar multiplication scales all values uniformly ( $kA = \text{all values scaled by k}$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Entry (2, 3) of matrix A + B represents the total iced coffee sold on Wednesday across both weeks, so entry (2, 3) = $25 + 15 = 40$ units, meaning the combined sales for that drink and day. Choice A is correct because it correctly interprets the entry in context as total iced coffee on Wednesday across weeks. Choice B confuses rows, interpreting entry (2, 3) as hot coffee on Wednesday when row 2 actually represents iced coffee. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
A gym lists monthly membership prices (in dollars) for two plans at two locations. Rows represent plans (row 1 = Basic, row 2 = Premium) and columns represent locations (column 1 = North, column 2 = South). The price matrix is $P=\begin{bmatrix}30 & 25\\ 50 & 45\end{bmatrix}$ All prices increase by 10%. Based on the matrix, what does the matrix $1.1P$ represent?
Answer: The new monthly prices after a 10% increase, in dollars This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources (A + B = total from both sources), subtraction finds changes or differences (A - B = net change), scalar multiplication scales all values uniformly (kA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar 1.1 means increasing all prices by 10%, so the new matrix is 1.1P where each entry is multiplied: for example, row 1, column 1 becomes 1.1 × 30 = 33 dollars. Choice A is correct because it properly identifies the matrix operation needed and correctly interprets the entry in context. Choice C interprets the matrix operation incorrectly, claiming a decrease when 1.1P actually represents an increase. Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities. When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
A store sells 2 products (row 1 = Product 1, row 2 = Product 2) in 3 weeks (column 1 = Week 1, column 2 = Week 2, column 3 = Week 3). Quantities sold matrix $Q$ (in units) is $Q=\begin{bmatrix}10 & 15 & 20\\ 5 & 10 & 15\end{bmatrix}$ The price per unit for each week is given by the column vector $p$ (in dollars per unit): $p=\begin{bmatrix}2\\ 3\\ 4\end{bmatrix}$ Based on the matrices, what does the matrix product $Qp$ represent in this context?
Answer: The total revenue for each product across all three weeks, in dollars This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B = \text{total from both sources}$ ), subtraction finds changes or differences ( $A - B = \text{net change}$ ), scalar multiplication scales all values uniformly ( $kA = \text{all values scaled by k}$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The product Qp combines the relationships: rows of Q (products) and column of p (weekly prices) relate, and entry (i,1) of Qp represents total revenue for product i, calculated as row-dot-column example for product 1: $10 \times 2 + 15 \times 3 + 20 \times 4 = 145$ dollars. Choice A is correct because it correctly interprets the entry in context. Choice B interprets the matrix operation incorrectly, claiming units sold when Qp actually represents revenue. For matrix multiplication in data contexts, think about how dimensions relate: if A is products×stores (showing inventory) and B is stores×warehouses (showing shipments), then AB is products×warehouses (showing total by product and destination). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities.
A farm ships apples and oranges to two markets. Rows represent products (row 1 = Apples, row 2 = Oranges) and columns represent markets (column 1 = Market A, column 2 = Market B). The shipment matrix $S$ (in crates) is $S=\begin{bmatrix}15 & 20\\ 10 & 25\end{bmatrix}$ If each crate sells for $2 at both markets, which calculation gives the revenue matrix (in dollars) for each product and market?
Answer: $2S$ This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $(A + B) = \text{total from both sources}$ ), subtraction finds changes or differences ( $(A - B) = \text{net change}$ ), scalar multiplication scales all values uniformly ( $(kA) = \text{all values scaled by } k$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar 2 means scaling shipments to revenue at $2 per crate, so the new matrix is $2S$ where each entry is multiplied: for example, row 1, column 1 becomes $2 \times 15 = 30$ dollars. Choice B is correct because it properly identifies the matrix operation needed. Choice A applies the percentage incorrectly, adding 2 instead of multiplying by 2. Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities. When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25$ to show you understand what the numbers represent.
A bookstore records the number of copies of 3 genres sold at 2 branches. Rows represent genres (row 1 = Fiction, row 2 = Nonfiction, row 3 = Comics) and columns represent branches (column 1 = East, column 2 = West). Monday sales matrix $M$ (in books) is $M=\begin{bmatrix}10 & 15\\ 20 & 10\\ 5 & 25\end{bmatrix}$ Tuesday sales matrix $T$ (in books) is $T=\begin{bmatrix}15 & 10\\ 10 & 15\\ 20 & 5\end{bmatrix}$ Using the sales data, what is the entry in row 3, column 1 of $M+T$ ?
Answer: 25 books This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B$ = total from both sources), subtraction finds changes or differences ( $A - B$ = net change), scalar multiplication scales all values uniformly ( $kA$ = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). To find total sales, we add matrices M and T: $(M + T)_{31} = M_{31} + T_{31}$ means we add corresponding entries, so $5 + 20 = 25$ , representing total comics sold at the East branch over both days in books. Choice A is correct because it accurately calculates and interprets the result. Choice C makes an arithmetic error, calculating $5 + 5 = 10$ instead of $5 + 20 = 25$ . Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
Two delivery services report the number of packages delivered (in packages) for two neighborhoods across three days. Rows represent neighborhoods (North, South) and columns represent days (Mon, Tue, Wed). Matrices $S_1$ and $S_2$ are: $S_1=\begin{bmatrix}20&25&30\\15&20&25\end{bmatrix},\quad S_2=\begin{bmatrix}10&5&15\\5&10&5\end{bmatrix}.$ Based on the matrices, what is the entry in row 1, column 2 of $S_1+S_2$ (with units)?
Answer: $30$ packages This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources (A + B = total from both sources), subtraction finds changes or differences (A - B = net change), scalar multiplication scales all values uniformly (kA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). To find total packages delivered, we add matrices S1 and S2: $(S_1 + S_2)_{12} = S_{1_{12}} + S_{2_{12}}$ means we add corresponding entries, so $25 + 5 = 30$ , representing total packages in North neighborhood on Tuesday. Choice B is correct because it accurately calculates and interprets the result with the correct value and units. Choice A makes an arithmetic error, calculating $20 + 5 = 25$ instead of $25 + 5 = 30$ . Remember that matrix operations preserve structure: if your original matrix is products×stores, then adding or subtracting another products×stores matrix keeps the same structure, and each entry in the result still represents the same type of information (just combined or differenced). When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
A small factory makes two products using two machines. Matrix $Q$ gives quantities produced (in units), where rows are products (Product 1, Product 2) and columns are machines (Machine A, Machine B): $Q=\begin{bmatrix}10&20\\15&5\end{bmatrix}.$ Matrix $C$ gives cost per unit (in dollars) for each machine, where rows are machines (Machine A, Machine B) and columns are cost categories (Labor, Materials): $C=\begin{bmatrix}5&10\\10&5\end{bmatrix}.$ Based on the matrices, what does the matrix product $QC$ represent in this context?
Answer: Total cost (in dollars) by product and cost category This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B$ = total from both sources), subtraction finds changes or differences ( $A - B$ = net change), scalar multiplication scales all values uniformly ( $kA$ = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The product QC combines the relationships: rows of Q (products) and columns of C (cost categories) relate through machines, and entry (i, j) of QC represents total cost for product i in cost category j, calculated as row-column dot product example for (1,1): $10 \times 5 + 20 \times 10 = 250$ dollars. Choice A is correct because it correctly interprets the entry in context after matrix multiplication. Choice C misinterprets the entry, stating it represents cost per unit when actually it represents total cost based on quantities and per-unit costs. For matrix multiplication in data contexts, think about how dimensions relate: if A is products $\times$ stores (showing inventory) and B is stores $\times$ warehouses (showing shipments), then $AB$ is products $\times$ warehouses (showing total by product and destination). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities.
A movie theater tracks ticket sales (in tickets) for three movies across two showtimes. Rows represent movies (Movie 1, Movie 2, Movie 3) and columns represent showtimes (Matinee, Evening). The sales matrix is $T=\begin{bmatrix}30&20\\25&30\\10&15\end{bmatrix}.$ Using the matrix, what is the total number of tickets sold for Movie 2 across both showtimes?
Answer: $55$ tickets This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrices provide a compact way to organize related data: rows and columns represent different categories (like products and stores, or students and test scores), and each entry (i, j) represents a specific data point (like the inventory of product i at store j). To find total tickets for Movie 2, we add entries in row 2: $25$ (Matinee) + $30$ (Evening) = $55$ tickets, representing combined sales across showtimes. Choice A is correct because it accurately calculates and interprets the sum with the correct value and units. Choice B confuses rows and columns, using row 1 column 2 ( $20$ ) + row 2 column 2 ( $30$ ) = $50$ instead of summing across row 2. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is $25$ ,' say 'the remaining inventory is $25$ units' or 'the new price is $25$ ' to show you understand what the numbers represent.
Two stores track weekly sales revenue (in dollars) for two products (P1, P2) across two days (Saturday, Sunday). Rows represent products and columns represent days. Store 1 has $S_1=\begin{bmatrix}100&150\\200&250\end{bmatrix}$ and Store 2 has $S_2=\begin{bmatrix}150&100\\250&200\end{bmatrix}.$ Using the sales data, what is the entry in row 2, column 2 of $S_1+S_2$ ?
Answer: $450 This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B = \text{total from both sources}$ ), subtraction finds changes or differences ( $A - B = \text{net change}$ ), scalar multiplication scales all values uniformly ( $kA = \text{all values scaled by } k$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). To find total revenue, we add matrices $S_1$ and $S_2$ : $(S_1 + S_2)_{22} = 250 + 200 = 450$ , representing the total revenue for P2 on Sunday across both stores. Choice A is correct because it accurately calculates and interprets the result for the specified entry. Choice B makes an arithmetic error, calculating perhaps $250 - 200 = 50$ instead of $250 + 200 = 450$ . Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.
A movie theater records the number of tickets sold (in people) for three movies (M1, M2, M3) across two showtimes (Early, Late). The ticket sales matrix is $T=\begin{bmatrix}50&60\\40&55\\30&45\end{bmatrix}.$ A promotion doubles attendance for every movie and showtime. Based on the matrix, what does the matrix $2T$ represent in this context?
Answer: The ticket sales after the promotion, with every entry (movie, showtime) doubled in people This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B = \text{total from both sources}$ ), subtraction finds changes or differences ( $A - B = \text{net change}$ ), scalar multiplication scales all values uniformly ( $kA = \text{all values scaled by } k$ ), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar 2 means doubling all ticket sales, so the new matrix is 2T where each entry is multiplied: for example, row 1, column 1 gives $2 \times 50 = 100$ people, representing doubled attendance for M1 at the Early showtime. Choice B is correct because it accurately describes 2T as the ticket sales after the promotion doubles every entry. Choice D interprets the scalar incorrectly, claiming adding 2 people when the context requires multiplying by 2 for doubling. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). Common applications: addition for totals from multiple sources, subtraction for net changes (before minus after, current minus previous), scalar multiplication for percentage changes (multiply by 1.2 for 20% increase), and matrix multiplication for combining unit costs with quantities.
A social network can be represented by an adjacency matrix $A$ where $A_{ij} = 1$ if person $i$ follows person $j$ , and $A_{ij} = 0$ otherwise. If $A^2$ represents the square of this matrix, what does the entry $(A^2)_{ij}$ tell us about the relationship between person $i$ and person $j$ ?
Answer: The number of two-step paths from person $i$ to person $j$ through intermediaries When we compute $A^2$ , the entry $(A^2)_{ij} = \sum_{k} A_{ik} \cdot A_{kj}$ . This counts how many intermediate people $k$ exist such that person $i$ follows person $k$ AND person $k$ follows person $j$ , which represents two-step paths. Choice A describes the dot product of row $i$ and row $j$ of $A$ , not $(A^2)_{ij}$ . Choice C would require comparing $A_{ij}$ and $A_{ji}$ . Choice D would be the sum of column $j$ in $A$ , not an entry in $A^2$ .
A delivery company tracks miles traveled on two routes over three days. Rows represent routes (row 1 = Route A, row 2 = Route B) and columns represent days (column 1 = Mon, column 2 = Tue, column 3 = Wed). Last week's miles matrix $L$ is $L=\begin{bmatrix}20 & 25 & 30\\ 15 & 20 & 25\end{bmatrix}$ This week's miles matrix $W$ is $W=\begin{bmatrix}25 & 20 & 30\\ 10 & 25 & 30\end{bmatrix}$ Using the matrices, what does entry $(2,3)$ of the matrix $W-L$ represent?
Answer: The change in miles on Route B on Wednesday from last week to this week This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources (A + B = total from both sources), subtraction finds changes or differences (A - B = net change), scalar multiplication scales all values uniformly (kA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The matrix W - L represents change in miles from last week to this week, calculated entry-by-entry: row 2, column 3 gives 30 - 25 = 5 miles, meaning the increase on Route B on Wednesday. Choice B is correct because it correctly interprets the entry in context. Choice D confuses rows and columns, interpreting entry (2,3) as representing Route A this week minus Route B last week instead of Route B Wednesday change. Key to matrix data problems: first understand what each row and column represents, then determine what each entry (i, j) means in context, and finally choose the matrix operation that makes sense for what you're trying to find (addition for combining, subtraction for changes, scalar multiplication for uniform scaling). Remember that matrix operations preserve structure: if your original matrix is products×stores, then adding or subtracting another products×stores matrix keeps the same structure, and each entry in the result still represents the same type of information (just combined or differenced).
A company tracks hours worked by 3 employees on 2 tasks. Rows represent employees (row 1 = Employee 1, row 2 = Employee 2, row 3 = Employee 3) and columns represent tasks (column 1 = Task A, column 2 = Task B). Hours matrix $H$ (in hours) is $H=\begin{bmatrix}5 & 10\\ 15 & 5\\ 10 & 15\end{bmatrix}$ Hourly wage for each task is given by $w=\begin{bmatrix}20\\ 30\end{bmatrix}$ Using the matrices, what is the entry in row 2 of the resulting vector $Hw$ ?
Answer: $450 This question tests understanding of how matrices can represent and manipulate data in real-world contexts. Matrix operations have practical interpretations: addition combines data from multiple sources ( $A + B =$ total from both sources), subtraction finds changes or differences ( $A - B =$ net change), scalar multiplication scales all values uniformly ( $kA =$ all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The product Hw combines the relationships: rows of H (employees) and column of w (wages) relate, and entry (2,1) of Hw represents total pay for employee 2, calculated as row-dot-column: $15\times20 + 5\times30 = 300 + 150 = 450$ dollars. Choice A is correct because it accurately calculates and interprets the result. Choice B makes an arithmetic error, calculating $15\times20 = 300$ instead of including both tasks. For matrix multiplication in data contexts, think about how dimensions relate: if A is products\times stores (showing inventory) and B is stores\times warehouses (showing shipments), then AB is products\times warehouses (showing total by product and destination). When interpreting results, always include units and context: don't just say 'the answer is 25,' say 'the remaining inventory is 25 units' or 'the new price is $25' to show you understand what the numbers represent.

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