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Precalculus Quiz

Precalculus Quiz: Using Matrices To Represent Manipulate Data

Practice Using Matrices To Represent Manipulate Data in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

A store's inventory system uses matrix III to track three items across four locations, and matrix CCC to track the cost per unit of each item. If III is 4×34 \times 34×3 and CCC is 3×13 \times 13×1, which matrix operation gives the total inventory value at each location?

Select an answer to continue

What this quiz covers

This quiz focuses on Using Matrices To Represent Manipulate Data, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A store's inventory system uses matrix III to track three items across four locations, and matrix CCC to track the cost per unit of each item. If III is 4×34 \times 34×3 and CCC is 3×13 \times 13×1, which matrix operation gives the total inventory value at each location?

  1. CT⋅ITC^T \cdot I^TCT⋅IT resulting in a 1×41 \times 41×4 matrix
  2. I⋅CI \cdot CI⋅C resulting in a 4×14 \times 14×1 matrix (correct answer)
  3. C⋅IC \cdot IC⋅I resulting in a 3×43 \times 43×4 matrix
  4. IT⋅CI^T \cdot CIT⋅C resulting in a 3×13 \times 13×1 matrix

Explanation: To find the total value at each location, we multiply the quantity of each item (from matrix III) by its cost (from matrix CCC) and sum across all items. This requires I⋅CI \cdot CI⋅C: a 4×34 \times 34×3 matrix times a 3×13 \times 13×1 matrix, giving a 4×14 \times 14×1 result where each entry represents the total inventory value at one location. Choice A gives total value per item across all locations. Choice C has incompatible dimensions. Choice D gives total quantity of each item across all locations multiplied by cost.

Question 2

A tournament bracket can be represented using an incidence matrix MMM where rows represent teams and columns represent matches. Entry Mij=1M_{ij} = 1Mij​=1 if team iii plays in match jjj, and Mij=0M_{ij} = 0Mij​=0 otherwise. If we want to find which teams play against each other in any match, we should examine:

  1. The matrix M⋅MTM \cdot M^TM⋅MT and look for entries equal to 1
  2. The matrix MT⋅MM^T \cdot MMT⋅M and look for off-diagonal entries equal to 1
  3. The matrix M+MTM + M^TM+MT and look for entries equal to 2
  4. The original matrix MMM and identify columns with exactly two 1's (correct answer)

Explanation: In an incidence matrix for tournament matches, each match involves exactly two teams, so each column should have exactly two 1's. Teams that play each other are those with 1's in the same column. Choice A: MMTMM^TMMT gives the number of matches each pair of teams shares (diagonal = total matches per team). Choice B: MTMM^TMMTM gives relationships between matches. Choice C: M+MTM + M^TM+MT doesn't provide meaningful information for this rectangular matrix structure.

Question 3

A small café tracks how many sandwiches and salads it has in stock at two locations. Rows represent items (row 1 = Sandwiches, row 2 = Salads) and columns represent locations (column 1 = Downtown, column 2 = Uptown). Inventory matrix AAA (in units) is A=[20301525]A=\begin{bmatrix}20 & 30\\ 15 & 25\end{bmatrix}A=[2015​3025​] Units sold during the day are in matrix BBB (in units): B=[5101015]B=\begin{bmatrix}5 & 10\\ 10 & 15\end{bmatrix}B=[510​1015​] Based on the matrices, what does the matrix A−BA-BA−B represent in this context?

  1. The total inventory across both locations after combining stock and sales, in units
  2. The remaining inventory at each location after the day’s sales, in units (correct answer)
  3. The total number of items sold at each location during the day, in units
  4. The inventory increase from the start of the day to the end of the day, in units

Explanation: 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 remaining inventory after sales, calculated entry-by-entry: for example, row 1, column 1 gives 20 - 5 = 15 units, meaning the sandwiches left downtown after sales. Choice B is correct because it properly identifies the matrix operation needed and correctly interprets the entry in context. Choice A suggests using addition when the context requires subtraction: to find remaining inventory, we need to subtract sales from starting stock, not add them. 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).

Question 4

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 QQQ (in units) is Q=[10152051015]Q=\begin{bmatrix}10 & 15 & 20\\ 5 & 10 & 15\end{bmatrix}Q=[105​1510​2015​] The price per unit for each week is given by the column vector ppp (in dollars per unit): p=[234]p=\begin{bmatrix}2\\ 3\\ 4\end{bmatrix}p=​234​​ Based on the matrices, what does the matrix product QpQpQp represent in this context?

  1. The total revenue for each product across all three weeks, in dollars (correct answer)
  2. The total units sold for each product across all three weeks, in units
  3. The revenue earned each week across both products, in dollars
  4. The price per unit for each product, in dollars per unit

Explanation: 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 sourcesA + B = \text{total from both sources}A+B=total from both sources), subtraction finds changes or differences (A−B=net changeA - B = \text{net change}A−B=net change), scalar multiplication scales all values uniformly (kA=all values scaled by kkA = \text{all values scaled by k}kA=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: 10×2+15×3+20×4=14510 \times 2 + 15 \times 3 + 20 \times 4 = 14510×2+15×3+20×4=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.

Question 5

A small café tracks how many sandwiches and salads it has in stock at two locations. Rows represent items (row 1 = Sandwiches, row 2 = Salads) and columns represent locations (column 1 = Downtown, column 2 = Uptown). Inventory matrix AAA (in units) is A=[20301525]A=\begin{bmatrix}20 & 30\\ 15 & 25\end{bmatrix}A=[2015​3025​] Units sold during the day are in matrix BBB (in units): B=[5101015]B=\begin{bmatrix}5 & 10\\ 10 & 15\end{bmatrix}B=[510​1015​] Based on the matrices, what does the matrix A−BA-BA−B represent in this context?​

  1. The total inventory across both locations after combining stock and sales, in units
  2. The remaining inventory at each location after the day’s sales, in units (correct answer)
  3. The total number of items sold at each location during the day, in units
  4. The inventory increase from the start of the day to the end of the day, in units

Explanation: 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 remaining inventory after sales, calculated entry-by-entry: for example, row 1, column 1 gives 20 - 5 = 15 units, meaning the sandwiches left downtown after sales. Choice B is correct because it properly identifies the matrix operation needed and correctly interprets the entry in context. Choice A suggests using addition when the context requires subtraction: to find remaining inventory, we need to subtract sales from starting stock, not add them. 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).

Question 6

A logistics company tracks delivery routes between cities using a connectivity matrix RRR where Rij=1R_{ij} = 1Rij​=1 if there's a direct route from city iii to city jjj, and Rij=0R_{ij} = 0Rij​=0 otherwise. The company also maintains a cost matrix KKK where KijK_{ij}Kij​ represents the cost of shipping from city iii to city jjj (with Kij=0K_{ij} = 0Kij​=0 if no direct route exists).

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?

  1. Calculate R⋅1R \cdot \mathbf{1}R⋅1 where 1\mathbf{1}1 is a column vector of ones (correct answer)
  2. Calculate the trace of matrix RRR for each row individually
  3. Calculate 1T⋅R\mathbf{1}^T \cdot R1T⋅R where 1T\mathbf{1}^T1T is a row vector of ones
  4. Calculate the element-wise product R∘1R \circ \mathbf{1}R∘1 where 1\mathbf{1}1 is a matrix of ones

Explanation: To count reachable cities from each city, we sum each row of matrix RRR. This is accomplished by multiplying RRR by a column vector of ones, giving R⋅1R \cdot \mathbf{1}R⋅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 RRR since multiplying by ones element-wise doesn't change values.

Question 7

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 SSS (in crates) is S=[15201025]S=\begin{bmatrix}15 & 20\\ 10 & 25\end{bmatrix}S=[1510​2025​] If each crate sells for \2$ at both markets, which calculation gives the revenue matrix (in dollars) for each product and market?

  1. S+2S+2S+2
  2. 2S2S2S (correct answer)
  3. S2S^2S2
  4. S−2S-2S−2

Explanation: 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(A + B) = \text{total from both sources}(A+B)=total from both sources), subtraction finds changes or differences ((A−B)=net change(A - B) = \text{net change}(A−B)=net change), scalar multiplication scales all values uniformly ((kA)=all values scaled by k(kA) = \text{all values scaled by } k(kA)=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 2percrate,sothenewmatrixis2 per crate, so the new matrix is 2percrate,sothenewmatrixis2Swhereeachentryismultiplied:forexample,row1,column1becomeswhere each entry is multiplied: for example, row 1, column 1 becomeswhereeachentryismultiplied:forexample,row1,column1becomes2 \times 15 = 30dollars.ChoiceBiscorrectbecauseitproperlyidentifiesthematrixoperationneeded.ChoiceAappliesthepercentageincorrectly,adding2insteadofmultiplyingby2.Commonapplications:additionfortotalsfrommultiplesources,subtractionfornetchanges(beforeminusafter,currentminusprevious),scalarmultiplicationforpercentagechanges(multiplyby1.2for20dollars. 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 isdollars.ChoiceBiscorrectbecauseitproperlyidentifiesthematrixoperationneeded.ChoiceAappliesthepercentageincorrectly,adding2insteadofmultiplyingby2.Commonapplications:additionfortotalsfrommultiplesources,subtractionfornetchanges(beforeminusafter,currentminusprevious),scalarmultiplicationforpercentagechanges(multiplyby1.2for2025$ to show you understand what the numbers represent.

Question 8

A theater tracks tickets sold for two showtimes at two locations. Rows represent showtimes (row 1 = Matinee, row 2 = Evening) and columns represent locations (column 1 = Theater 1, column 2 = Theater 2). Online ticket sales matrix OOO (in tickets) is O=[25203015]O=\begin{bmatrix}25 & 20\\ 30 & 15\end{bmatrix}O=[2530​2015​] Box-office sales matrix BBB (in tickets) is B=[10152025]B=\begin{bmatrix}10 & 15\\ 20 & 25\end{bmatrix}B=[1020​1525​] Using the matrices, which operation finds the total tickets sold (online + box office) for each showtime and location?​

  1. OBOBOB
  2. O−BO-BO−B
  3. O+BO+BO+B (correct answer)
  4. 2O2O2O

Explanation: 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 tickets, we add matrices O and B: (O + B)ᵢⱼ = Oᵢⱼ + Bᵢⱼ means we add corresponding entries, so for example row 1, column 1 = 25 + 10 = 35 tickets, representing total matinee at Theater 1. Choice C is correct because it properly identifies the matrix operation needed. Choice B suggests using subtraction when the context requires addition: to find totals, we need to add sources, not subtract. 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).

Question 9

A bookstore tracks how many copies of each title are in stock at two locations. Rows represent titles (Title 1, Title 2, Title 3) and columns represent stores (Store A, Store B). Matrix AAA shows starting inventory (in units), and matrix BBB shows units sold during the week:

A=[203015251035],B=[510105515].A=\begin{bmatrix}20&30\\15&25\\10&35\end{bmatrix},\quad B=\begin{bmatrix}5&10\\10&5\\5&15\end{bmatrix}.A=​201510​302535​​,B=​5105​10515​​.

Based on the matrices, what does the matrix A−BA-BA−B represent in this context?

  1. The total inventory and sales combined (in units) for each title at each store
  2. The remaining inventory (in units) for each title at each store after the week’s sales (correct answer)
  3. The percent of inventory sold for each title at each store
  4. The number of stores that carry each title

Explanation: 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+BA + BA+B = total from both sources), subtraction finds changes or differences (A−BA - BA−B = net change), scalar multiplication scales all values uniformly (kAkAkA = all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). The matrix A−BA - BA−B represents remaining inventory after sales, calculated entry-by-entry: for example, row 1 column 1 gives 20−5=1520 - 5 = 1520−5=15 units, meaning the remaining copies of Title 1 at Store A after sales. Choice B is correct because it properly identifies the matrix operation needed and correctly interprets the result as the remaining inventory for each title at each store. Choice A suggests using addition when the context requires subtraction: to find remaining inventory, we need to subtract sales from starting inventory, not add them. 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).

Question 10

A city transit planner uses an adjacency matrix MMM to represent direct train connections between three stations (A, B, C). Entry (i,j)(i,j)(i,j) is 1 if there is a direct train from station iii to station jjj, and 0 otherwise: M=[011100010].M=\begin{bmatrix}0&1&1\\1&0&0\\0&1&0\end{bmatrix}.M=​010​101​100​​. For the network described, what does the matrix M2M^2M2 represent in this context?

  1. The number of 2-step routes between stations, where entry (i,j)(i,j)(i,j) counts how many ways to go from iii to jjj using exactly two trains (correct answer)
  2. The number of direct connections between stations, where entry (i,j)(i,j)(i,j) is 1 if a direct train exists
  3. The total number of stations reachable from each station in one step, listed as a single number
  4. The distance (in miles) between each pair of stations

Explanation: 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+BA + BA+B = total from both sources), subtraction finds changes or differences (A−BA - BA−B = net change), scalar multiplication scales all values uniformly (kAkAkA = 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: each entry (i, j) of M² represents the number of 2-step paths from station i to station j, calculated as the dot product of row i and column j of M, for example, entry (1,1) = 0×0+1×1+1×0=10\times0 + 1\times1 + 1\times0 = 10×0+1×1+1×0=1 path. Choice A is correct because it properly identifies the matrix operation needed and correctly interprets M² as counting 2-step routes between stations. Choice B confuses M² with the original M, stating it represents direct connections when M² actually represents indirect 2-step connections. 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.

Question 11

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=[30255045]P=\begin{bmatrix}30 & 25\\ 50 & 45\end{bmatrix}P=[3050​2545​] All prices increase by 10%. Based on the matrix, what does the matrix 1.1P1.1P1.1P represent?

  1. The new monthly prices after a 10% increase, in dollars (correct answer)
  2. The amount of the increase only (not including original prices), in dollars
  3. The original monthly prices decreased by 10%, in dollars
  4. The total number of memberships sold after a 10% increase, in memberships

Explanation: 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 sourcesA + B = \text{total from both sources}A+B=total from both sources), subtraction finds changes or differences (A−B=net changeA - B = \text{net change}A−B=net change), scalar multiplication scales all values uniformly (kA=all values scaled by kkA = \text{all values scaled by } kkA=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=331.1 \times 30 = 331.1×30=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.

Question 12

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 MMM (in books) is M=[10152010525]M=\begin{bmatrix}10 & 15\\ 20 & 10\\ 5 & 25\end{bmatrix}M=​10205​151025​​ Tuesday sales matrix TTT (in books) is T=[15101015205]T=\begin{bmatrix}15 & 10\\ 10 & 15\\ 20 & 5\end{bmatrix}T=​151020​10155​​ Using the sales data, what is the entry in row 3, column 1 of M+TM+TM+T?​

  1. 25 books (correct answer)
  2. 20 books
  3. 10 books
  4. 30 books

Explanation: 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 = M₃₁ + T₃₁ 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.

Question 13

A store tracks inventory (in units) of two products (P1, P2) at three locations (L1, L2, L3). Matrix AAA is last month’s inventory and matrix BBB is this month’s inventory: A=[202530152025],B=[252030102520].A=\begin{bmatrix}20&25&30\\15&20&25\end{bmatrix},\quad B=\begin{bmatrix}25&20&30\\10&25&20\end{bmatrix}.A=[2015​2520​3025​],B=[2510​2025​3020​]. Which matrix operation finds the net change in inventory from last month to this month (this month minus last month) for each product at each location?

  1. A+BA+BA+B
  2. A−BA-BA−B
  3. B−AB-AB−A (correct answer)
  4. ABABAB

Explanation: 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 B - A represents the net change in inventory (this month minus last month), calculated entry-by-entry: for example, row 1, column 1 gives 25 - 20 = 5 units, meaning the net increase for P1 at L1. Choice C is correct because it properly identifies the matrix operation needed to find the net change as B - A. Choice B suggests using A - B, which would give the negative of the net change, representing last month minus this month instead of the desired direction. 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.

Question 14

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=[551051010].P=\begin{bmatrix}5&5&10\\5&10&10\end{bmatrix}.P=[55​510​1010​]. All prices increase by 10%. Based on the matrix, which calculation gives the new price matrix?

  1. P+0.10P+0.10P+0.10
  2. 1.10P1.10P1.10P (correct answer)
  3. P−0.10PP-0.10PP−0.10P
  4. P2P^{2}P2

Explanation: 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 sourcesA + B = \text{total from both sources}A+B=total from both sources), subtraction finds changes or differences (A−B=net changeA - B = \text{net change}A−B=net change), scalar multiplication scales all values uniformly (kA=all values scaled by kkA = \text{all values scaled by } kkA=all values scaled by k), and matrix multiplication combines relationships (like quantities times unit costs to get total costs). Multiplying by scalar 1.101.101.10 means increasing all prices by 10%, so the new matrix is 1.10P1.10P1.10P where each entry is multiplied: for example, row 1 column 1 becomes 1.10×5=5.501.10 \times 5 = 5.501.10×5=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+0.101 + 0.101+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 1.21.21.2 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.

Question 15

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 S1S_1S1​ and S2S_2S2​ are:

S1=[202530152025],S2=[105155105].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}.S1​=[2015​2520​3025​],S2​=[105​510​155​].

Based on the matrices, what is the entry in row 1, column 2 of S1+S2S_1+S_2S1​+S2​ (with units)?

  1. 252525 packages
  2. 303030 packages (correct answer)
  3. 353535 packages
  4. 202020 packages

Explanation: 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: (S1+S2)12=S112+S212(S_1 + S_2)_{12} = S_{1_{12}} + S_{2_{12}}(S1​+S2​)12​=S112​​+S212​​ means we add corresponding entries, so 25+5=3025 + 5 = 3025+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=2520 + 5 = 2520+5=25 instead of 25+5=3025 + 5 = 3025+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.

Question 16

A manufacturing company produces three products using four different machines. The production matrix MMM shows hours needed per unit, where MijM_{ij}Mij​ represents hours that machine iii needs to produce one unit of product jjj. If the company receives an order represented by vector v=[50,30,40]T\mathbf{v} = [50, 30, 40]^Tv=[50,30,40]T (units of each product), which calculation determines the total hours each machine will operate?

  1. vT⋅M\mathbf{v}^T \cdot MvT⋅M resulting in a 1×41 \times 41×4 vector
  2. M⋅vM \cdot \mathbf{v}M⋅v resulting in a 4×14 \times 14×1 vector (correct answer)
  3. MT⋅vM^T \cdot \mathbf{v}MT⋅v resulting in a 3×13 \times 13×1 vector
  4. v⋅MT\mathbf{v} \cdot M^Tv⋅MT resulting in a 3×43 \times 43×4 matrix

Explanation: To find total hours per machine, we multiply the hours needed per unit (matrix MMM) by the number of units ordered (vector v\mathbf{v}v). Since MMM is 4×34 \times 34×3 and v\mathbf{v}v is 3×13 \times 13×1, the operation M⋅vM \cdot \mathbf{v}M⋅v gives a 4×14 \times 14×1 vector where each entry represents total operating hours for one machine. Choice A has incompatible dimensions. Choice C would give total hours needed per product type across all machines. Choice D also has incompatible dimensions for the stated operation.

Question 17

A social network can be represented by an adjacency matrix AAA where Aij=1A_{ij} = 1Aij​=1 if person iii follows person jjj, and Aij=0A_{ij} = 0Aij​=0 otherwise. If A2A^2A2 represents the square of this matrix, what does the entry (A2)ij(A^2)_{ij}(A2)ij​ tell us about the relationship between person iii and person jjj?

  1. The number of people that both person iii and person jjj follow
  2. The number of two-step paths from person iii to person jjj through intermediaries (correct answer)
  3. Whether person iii and person jjj follow each other mutually
  4. The total number of followers that person jjj has in the network

Explanation: When we compute A2A^2A2, the entry (A2)ij=∑kAik⋅Akj(A^2)_{ij} = \sum_{k} A_{ik} \cdot A_{kj}(A2)ij​=∑k​Aik​⋅Akj​. This counts how many intermediate people kkk exist such that person iii follows person kkk AND person kkk follows person jjj, which represents two-step paths. Choice A describes the dot product of row iii and row jjj of AAA, not (A2)ij(A^2)_{ij}(A2)ij​. Choice C would require comparing AijA_{ij}Aij​ and AjiA_{ji}Aji​. Choice D would be the sum of column jjj in AAA, not an entry in A2A^2A2.

Question 18

A research study tracks the interaction frequency between different animal species in an ecosystem using matrix SSS. If SijS_{ij}Sij​ represents the number of observed interactions between species iii and species jjj, and the matrix is symmetric (Sij=SjiS_{ij} = S_{ji}Sij​=Sji​), what does the sum of all entries in row iii represent?

  1. The total number of interactions involving species iii with all other species (correct answer)
  2. The total number of species that interact with species iii
  3. The total number of interactions between species iii and itself only
  4. The average interaction frequency for species iii across all other species

Explanation: Since each entry SijS_{ij}Sij​ represents the number of interactions between species iii and species jjj, summing all entries in row iii gives the total number of interactions involving species iii. This includes interactions with all other species plus self-interactions (SiiS_{ii}Sii​). Choice B would count non-zero entries, not sum them. Choice C refers only to the diagonal entry SiiS_{ii}Sii​. Choice D would require dividing the sum by the number of other species.

Question 19

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=[302025301015].T=\begin{bmatrix}30&20\\25&30\\10&15\end{bmatrix}.T=​302510​203015​​.

Using the matrix, what is the total number of tickets sold for Movie 2 across both showtimes?

  1. 555555 tickets (correct answer)
  2. 303030 tickets
  3. 252525 tickets
  4. 454545 tickets

Explanation: 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: 252525 (Matinee) + 303030 (Evening) = 555555 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 (202020) + row 2 column 2 (303030) = 505050 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 252525,' say 'the remaining inventory is 252525 units' or 'the new price is 252525' to show you understand what the numbers represent.

Question 20

A small business tracks its quarterly expenses across three categories: Labor, Materials, and Overhead. The data for the past year is represented in matrix EEE, where each row represents a quarter and each column represents a category. If the business wants to calculate the total annual expense for each category, which matrix operation should be performed?

  1. Calculate the transpose of matrix EEE and then sum each row
  2. Multiply matrix EEE by a 1×31 \times 31×3 matrix of ones
  3. Multiply a 1×41 \times 41×4 matrix of ones by matrix EEE (correct answer)
  4. Calculate the determinant of matrix EEE if it were square

Explanation: To find the total annual expense for each category, we need to sum the values in each column of the 4×34 \times 34×3 matrix EEE. Multiplying a 1×41 \times 41×4 matrix of ones [1,1,1,1][1, 1, 1, 1][1,1,1,1] by matrix EEE results in a 1×31 \times 31×3 matrix where each element is the sum of the corresponding column. Choice A would give row sums of the transposed matrix (quarter totals by category). Choice B has incompatible dimensions for multiplication. Choice D is irrelevant since the matrix isn't square and determinants don't give category totals.