Matrices: Class 12 Maths Practice Questions

A school tuck shop records the number of sandwiches (S), rolls (R) and wraps (W) sold on Monday and Tuesday. Monday sales: M = [[12, 8, 5]], Tuesday sales: T = [[15, 6, 9]] (1×3 row matrices). Selling prices: sandwich ₹30, roll ₹20, wrap ₹25, stored as column matrix P = [[30], [20], [25]].

A school tuck shop records the number of sandwiches (S), rolls (R) and wraps (W) sold on Monday and Tuesday in a matrix. The sales matrix for Monday is M = [[12, 8, 5]] and for Tuesday is T = [[15, 6, 9]] (each a 1×3 row matrix representing [S, R, W]).

The selling prices (in ₹) of a sandwich, a roll and a wrap are ₹30, ₹20 and ₹25 respectively, stored in a column matrix P = [[30], [20], [25]].

(i) Write down the combined sales matrix C = M + T and state its order. [1]
(ii) Using matrix multiplication, find the total revenue collected on Monday alone. [1]
(iii) Find the total revenue collected over both days combined, using the matrix C found in part (i). [2]

A school's science club tracks the performance of its three teams — Alpha, Beta, and Gamma — over two consecutive rounds of an inter-school quiz. The scores in Round 1 and Round 2 are recorded in matrices P and Q respectively:

P = [[4, 6, 2], [3, 5, 7]], Q = [[1, 3, 5], [6, 2, 4]]

where rows represent the two subjects (Physics and Chemistry) and columns represent the three teams (Alpha, Beta, Gamma).

A school's science club tracks the performance of its three teams — Alpha, Beta, and Gamma — over two consecutive rounds of an inter-school quiz. The scores in Round 1 and Round 2 are recorded in matrices P and Q respectively:

P = [[4, 6, 2], [3, 5, 7]], Q = [[1, 3, 5], [6, 2, 4]]

where rows represent the two subjects (Physics and Chemistry) and columns represent the three teams (Alpha, Beta, Gamma).

(i) Find the matrix representing the total scores of each team in each subject over both rounds. [1]
(ii) The club decides to give double weightage to Round 2 scores. Find the matrix representing the combined weighted score (Round 1 + 2 × Round 2). [1]
(iii) Using your answer from part (ii), find the matrix S = P + 2Q − 3I₂, where I₂ is the 2×2 identity matrix. What difficulty does this operation encounter, and why? [2]

A school management committee is preparing a budget report. The monthly expenditure (in ₹ thousands) on three heads — Salaries, Infrastructure, and Miscellaneous — for two departments (Science and Arts) is recorded in matrix A, and a 15% increase is proposed for the next month represented by B = 1.15A.

A = [[120, 80, 40], [90, 60, 30]]

A school management committee is preparing a budget report. The monthly expenditure (in ₹ thousands) on three heads — Salaries, Infrastructure, and Miscellaneous — for two departments (Science and Arts) is recorded in matrix A, and a 15% increase proposed for the next month is represented by matrix B = 1.15A.

The current monthly expenditure matrix is:

A = [[120, 80, 40], [90, 60, 30]]

where rows represent Science and Arts departments respectively, and columns represent Salaries, Infrastructure, and Miscellaneous.

Based on the above information, answer the following:

(i) Write the matrix B representing next month's proposed expenditure. [1]

(ii) The committee decides the total expenditure matrix for the two months combined is T = A + B. Find matrix T. [1]

(iii) If the committee wants to express T in the form T = kA for some scalar k, find the value of k. Hence verify that the total Salaries expenditure for the Science department obtained from T equals k times the corresponding entry in A. [2]

A manufacturing company tracks its monthly production and pricing data using matrices. The production matrix P (units produced) and the price matrix C (cost per unit in ₹) for two products X and Y across two plants are:

P = [[120, 80], [150, 60]] (rows = Plant 1, Plant 2; columns = Product X, Product Y)

C = [[200], [350]] (rows = Product X, Product Y)

The Revenue matrix R = PC, and the Bonus matrix B = P + Pᵀ.

A manufacturing company tracks its monthly production and pricing data using matrices. The production matrix P (units produced) and the price matrix C (cost per unit in ₹) for two products X and Y across two plants are given as:

P = [[120, 80], [150, 60]] where rows represent Plant 1 and Plant 2, columns represent Product X and Product Y.

C = [[200], [350]] where rows represent Product X and Product Y.

The Revenue matrix R is defined by R = PC, and a Bonus matrix B is defined such that B = P + Pᵀ (where Pᵀ is the transpose of P).

Based on this context, answer the following:

(i) Find the order of the Revenue matrix R = PC. Also compute R, interpreting each element in context. [2]

(ii) A quality analyst claims: 'The matrix B = P + Pᵀ is always a symmetric matrix for any matrix P.' Verify this claim analytically (without computing B explicitly), and state whether the analyst's claim is correct. [2]

A school's attendance monitoring system records daily presence of students in two sections, P and Q, over two consecutive days. The data is stored as matrices:

Day 1: A = [[30, 12], [8, 25]], Day 2: B = [[28, 15], [10, 22]]

Rows represent Sections P and Q; columns represent Boys and Girls present.

A school's attendance monitoring system records daily presence of students in two sections, P and Q, over two consecutive days. The data is stored as matrices:

Day 1: A = [[30, 12], [8, 25]] where rows represent Sections P and Q, and columns represent Boys and Girls present.

Day 2: B = [[28, 15], [10, 22]]

The school administration defines a 'Combined Attendance Matrix' C = A + B, and an 'Adjusted Matrix' D = 2A − B.

(i) Find the Combined Attendance Matrix C = A + B. [1 mark]
(ii) Find the Adjusted Matrix D = 2A − B. [1 mark]
(iii) The principal wants to award sections based on a 'Performance Score Matrix' S = C + D. Find S, and hence determine which section (P or Q) has the higher total attendance score (sum of Boys and Girls entries in that row of S). [2 marks]

A school tuck-shop records its daily sales of three items — Samosa (S), Sandwich (W) and Cold Drink (C) — for Monday and Tuesday in matrix form:

Monday: M = [[12, 8, 15], [10, 6, 20]]
Tuesday: T = [[14, 10, 12], [8, 9, 18]]

where Row 1 = Morning break sales and Row 2 = Lunch break sales.

The price (in ₹) of each item is given by the column matrix:

P = [[10], [15], [20]]

A school tuck-shop records its daily sales of three items — Samosa (S), Sandwich (W) and Cold Drink (C) — for Monday and Tuesday in matrix form:

Monday: M = [[12, 8, 15], [10, 6, 20]]
Tuesday: T = [[14, 10, 12], [8, 9, 18]]

where Row 1 = Morning break sales and Row 2 = Lunch break sales.

The price (in ₹) of each item is given by the column matrix:

P = [[10], [15], [20]]

(i) Write the matrix S = M + T that gives the total sales (item-wise, break-wise) over the two days. [1]
(ii) Find the total revenue collected during the Morning break over the two days. [1]
(iii) Find the matrix R = S × P giving the total revenue (in ₹) for each break over the two days, and hence state which break generated more revenue. [2]

A school tuck shop tracks daily sales of three items — Sandwich (S), Juice (J) and Cookie (C) — over two days. Day 1 sales: S = 40, J = 25, C = 60. Day 2 sales: S = 55, J = 30, C = 45. Selling price per unit (₹): S = 30, J = 20, C = 10. Cost price per unit (₹): S = 18, J = 12, C = 6.

A school tuck shop tracks daily sales of three items — Sandwich (S), Juice (J) and Cookie (C) — over two days using matrices.

Day 1 sales (units): S = 40, J = 25, C = 60
Day 2 sales (units): S = 55, J = 30, C = 45

The selling price (in ₹) per unit is: S = 30, J = 20, C = 10.
The cost price (in ₹) per unit is: S = 18, J = 12, C = 6.

Represent the sales data as a 2×3 matrix A, and the price data as a 3×2 matrix B where the first column gives selling prices and the second column gives cost prices of each item.

(i) Write down matrices A and B. [1]
(ii) Find the matrix P = AB and state what each element of P represents in context. [1]
(iii) Hence, find the matrix representing the profit earned (in ₹) on each day, and find the total profit over both days. [2]

A school is preparing merit certificates for three streams — Science, Commerce, and Arts. The number of Gold, Silver, and Bronze certificates awarded to each stream is represented by a 3×3 matrix A, and the cost of each certificate type is stored in a column matrix C. Matrix multiplication is used to compute stream-wise and overall expenditure.

A school is preparing merit certificates for students. The data for three categories — Science, Commerce, and Arts — is stored in a matrix. The number of students receiving Gold, Silver, and Bronze certificates in each stream is given by the matrix:

A = [[3, 5, 2], [4, 1, 6], [2, 3, 4]]

where rows represent Science, Commerce, and Arts respectively, and columns represent Gold, Silver, and Bronze respectively.

The cost (in ₹) of each type of certificate is stored as:
Gold = ₹150, Silver = ₹100, Bronze = ₹75

(i) Write the cost matrix C as a 3×1 column matrix. [1]
(ii) Find the product matrix AC, which gives the total certificate expenditure for each stream. [1]
(iii) The school decides to add a 'Special Merit' certificate costing ₹200 each. The number of Special Merit certificates for Science, Commerce, and Arts are 1, 2, and 3 respectively. Find the new total expenditure matrix and hence compute the grand total expenditure across all three streams. [2]

If A is a square matrix such that A² = A, then (I + A)² − 3A is equal to:

If A = [[k, 0], [0, k]] is a scalar matrix of order 2, then A² is equal to:

If A is any square matrix, then which of the following is always a symmetric matrix?

Assertion (A) : For any square matrix A, the matrix (A + A') is always a symmetric matrix.

Reason (R) : A matrix B is symmetric if and only if B' = B.

If A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, show that A² − 5A + 7I = O.

If A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, show that A² − 5A + 7I = O, where I is the identity matrix of order 2 and O is the zero matrix of order 2.

If $\left[\begin{array}{cc}3x-1& 4\\ 7& 2y+3\end{array}\right]=\left[\begin{array}{cc}8& 4\\ 7& 9\end{array}\right]$, find the value of $x+y$.

If A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}, find the matrix B such that A + B = 2I, where I is the identity matrix of order 2.

Express the matrix $A=\left[\begin{array}{ccc}3& -2& 1\\ 4& 0& -3\\ -1& 5& 2\end{array}\right]$ as the sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$. Write $P$ and $Q$ explicitly.

If A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} and f(x) = x² - 6x + 17, find f(A). Hence show that f(A) = 6I, where I is the identity matrix of order 2.

If A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}, verify that A² − 6A + 11I = O, where I is the 2×2 identity matrix and O is the 2×2 zero matrix.

If A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, show that A + A' = 2\cos\theta \cdot I, where I is the identity matrix of order 2. Hence find the value of \theta \in [0, 2\pi) for which A + A' = \sqrt{3}\, I.

If A = \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix}, show that A² − 5A + 7I = O, where I is the identity matrix of order 2 and O is the zero matrix of order 2. Hence find A⁻¹.

Let A = [aᵢⱼ] be a 3×3 matrix where aᵢⱼ = 2i − j. Express A as the sum of a symmetric matrix P and a skew-symmetric matrix Q. Write only the matrices P and Q explicitly.

A school library tracks books borrowed by Classes 10, 11, and 12 across Fiction, Science, and History categories over two weeks. The borrowing data is stored in 3×3 matrices W₁ and W₂. The library uses matrix operations to analyse borrowing patterns and award reward points.

A school library is tracking the number of books borrowed across three categories — Fiction (F), Science (S), and History (H) — for two consecutive weeks. The data is recorded in matrix form as follows:

Week 1: W₁ = [[12, 8, 5], [7, 15, 10], [9, 6, 14]]

Week 2: W₂ = [[10, 12, 7], [5, 9, 13], [11, 8, 6]]

where rows represent Classes 10, 11, and 12 respectively, and columns represent Fiction, Science, and History.

Based on this information, answer the following sub-questions:

(i) Find the total number of books borrowed by each class over both weeks (compute W₁ + W₂).

(ii) The librarian wants to find the difference in borrowing between Week 1 and Week 2, i.e., W₁ − W₂. Compute this matrix and identify which class borrowed MORE Science books in Week 1 compared to Week 2.

(iii) If the library assigns a reward-point matrix R = [[2], [3], [1]] (points per book for Fiction, Science, History respectively), find the total reward points earned by each class in Week 1 by computing W₁ · R. Also verify the order of the resulting matrix.

A school tuck-shop owner tracks sales of three snack items — Samosa (S), Burger (B), and Cold Drink (C) — over two days using matrices.

Day 1 sales (quantity): S = 40, B = 25, C = 30
Day 2 sales (quantity): S = 35, B = 30, C = 20

The price (in ₹) of each item is: S = 15, B = 30, C = 20
The cost price (in ₹) of each item is: S = 10, B = 20, C = 12

A school tuck-shop owner tracks sales of three snack items — Samosa (S), Burger (B), and Cold Drink (C) — over two days using matrices.

Day 1 sales (quantity): S = 40, B = 25, C = 30
Day 2 sales (quantity): S = 35, B = 30, C = 20

The price (in ₹) of each item is: S = 15, B = 30, C = 20
The cost price (in ₹) of each item is: S = 10, B = 20, C = 12

Represent the sales data as a 2×3 matrix Q, the selling price as a 3×1 column matrix P, and the cost price as a 3×1 column matrix C.

(i) Write down the matrices Q, P, and C. [1]
(ii) Compute the total revenue earned on each day using matrix multiplication. [1]
(iii) Find the total profit earned on each of the two days. Also find the total combined profit over both days. [2]

A school committee records student participation in Sports (S) and Cultural (C) activities over two terms. Matrix P gives the number of participants (rows = Terms, columns = Activities) and matrix Q gives the cost per student (in ₹) for each activity.

A school committee is preparing a report on two types of activities — Sports (S) and Cultural (C) — conducted over two terms (Term 1 and Term 2). The number of students who participated in each activity per term is recorded in a matrix P, and the cost (in ₹) per student for each activity is recorded in matrix Q, where:

P = [[40, 25], [50, 30]] (rows: Term 1, Term 2; columns: S, C)

Q = [[150], [200]] (rows: S, C; column: cost per student)

(i) Find the matrix PQ representing the total activity expenditure for each term.
(ii) Find the total expenditure incurred by the school across both terms.
(iii) The committee decides to give a 10% discount on Sports cost and a 5% discount on Cultural cost in Term 2 only. Write the revised cost matrix Q' for Term 2 and find the revised expenditure for Term 2.

A data scientist models a transformation on ℝ² using the matrix A = [[2, -1], [3, -2]]. During analysis, she observes that repeated application of this transformation has a special property.

A data scientist models a transformation on ℝ² using the matrix A = [[2, -1], [3, -2]]. During analysis, she observes that repeated application of this transformation has a special property.

(i) Compute A². What do you notice about A²? [1 mark]

(ii) Using the result of (i), find A¹⁰⁰. [1 mark]

(iii) If B = A² - 3A + 2I, find the matrix B. Hence determine whether B is singular or non-singular, justifying your answer. [2 marks]

A factory uses two types of raw materials, Material X and Material Y, to produce two products, Product P and Product Q. The quantity (in kg) of each material required per unit of each product is given by the matrix A = [[3, 1], [2, 4]], where rows represent Material X and Material Y respectively, and columns represent Product P and Product Q respectively. The factory receives an order where the total requirement is 11 kg of Material X and 22 kg of Material Y.

A factory uses two types of raw materials, Material X and Material Y, to produce two products, Product P and Product Q. The quantity (in kg) of each material required per unit of each product is given by the matrix:

A = [[3, 1], [2, 4]]

where rows represent Material X and Material Y respectively, and columns represent Product P and Product Q respectively.

The factory receives an order to produce a batch where the total requirement is 11 kg of Material X and 22 kg of Material Y.

(i) Write the system of linear equations representing the total material requirements if x units of Product P and y units of Product Q are to be produced. [1]
(ii) Express this system in the matrix form AX = B, identifying matrices A, X and B clearly. [1]
(iii) Find A⁻¹ and hence determine the number of units of Product P and Product Q to be produced to exactly meet the material requirements. [2]

A school treasurer models monthly fund allocations using a 2×2 matrix A = [[4, 6], [2, 5]], and needs to find individual donor contributions using the matrix equation AX = B, where B = [[26], [19]].

A school treasurer is managing two budget accounts — a Sports Fund and a Cultural Fund. The combined allocation matrix for two consecutive months is represented as:

A = [[4, 6], [2, 5]]

where the rows represent Month 1 and Month 2, and the columns represent Sports Fund (in ₹ thousands) and Cultural Fund (in ₹ thousands) respectively.

The treasurer needs to recover the original monthly contributions from two donors P and Q. The total contribution by donor P in Month 1 was ₹26 thousand and in Month 2 was ₹19 thousand. The contributions of each donor per fund per month are given by the matrix equation AX = B, where X = [[x], [y]] represents the per-unit contributions of donors P and Q, and B = [[26], [19]].

(i) Find the determinant of A and verify that A⁻¹ exists.
(ii) Find A⁻¹.
(iii) Using A⁻¹, solve for x and y, and interpret what x and y represent in this context.

A data analyst models a one-way communication network between three cities P, Q, R using a 3×3 matrix C = [[0,1,1],[1,0,1],[0,1,0]], where cᵢⱼ = 1 denotes a direct link from city i to city j. The matrix C² gives two-step connections and D = C + C² gives total connections up to two steps.

A data analyst is modelling a communication network between three cities P, Q, and R. The direct connections (one-way) between cities are represented by a 3×3 matrix C, where the entry cᵢⱼ = 1 if there is a direct link from city i to city j, and cᵢⱼ = 0 otherwise (with no self-loops). The analyst uses the matrix

C = [[0, 1, 1], [1, 0, 1], [0, 1, 0]]

where rows/columns 1, 2, 3 represent cities P, Q, R respectively.

The matrix C² gives the number of two-step connections between cities.

(i) Compute C² and interpret the entry (C²)₁₂ in the context of the problem. [1]

(ii) The analyst claims that the total number of ways to travel from city P to city R in exactly two steps equals (C²)₁₃. Verify this claim by listing all possible two-step routes from P to R, and confirm it matches (C²)₁₃. [1]

(iii) The matrix D = C + C² represents total connections (direct and two-step) between cities. Compute D and determine whether D is a symmetric matrix. If not, explain what the symmetry (or lack of it) tells us about the network. [2]

A school is organising a three-day Science Fair. Three stalls — Physics (P), Chemistry (C) and Biology (B) — sell entry tickets priced at ₹20, ₹30 and ₹25 respectively. The number of tickets sold on each day is recorded in matrix T (rows: Day 1, Day 2, Day 3; columns: Physics, Chemistry, Biology) and the price per ticket in ₹ is recorded in column matrix P (rows: Physics, Chemistry, Biology):

T = [[120, 80, 100], [150, 60, 90], [110, 95, 75]], P = [[20], [30], [25]]

A school is organising a three-day Science Fair. Three stalls — Physics (P), Chemistry (C) and Biology (B) — sell entry tickets priced at ₹20, ₹30 and ₹25 respectively.

The number of tickets sold on each day is recorded in matrix T and the price per ticket (in ₹) is recorded in matrix P:

T = [[120, 80, 100], [150, 60, 90], [110, 95, 75]] (rows: Day 1, Day 2, Day 3; columns: Physics, Chemistry, Biology)

P = [[20], [30], [25]] (rows: Physics, Chemistry, Biology)

Based on the above information, answer the following:

(i) Write the order of matrix T and matrix P. Hence state the order of the product matrix TP. [1 mark]

(ii) Compute the total revenue collected on Day 1 by evaluating the first row of the product TP. [1 mark]

(iii) Find the complete product matrix TP to determine the total revenue collected on each of the three days. Hence find the day on which the maximum revenue was collected. [2 marks]