CBSE Class 12 Mathematics Matrices and Determinants MCQs Set 06

Assertion and Reason Questions

Directions: In the context of above two statements, which one of the following is correct?
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : If \( A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix} \), then adj (adj A) = A.
Reason (R) : \( | \text{adj (adj A)} | = | A |^{(n-1)^2} \), A be n rowed non-singular matrix
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion (A) : If \( A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \), then \( A^{-1} = \begin{bmatrix} 1/a & 0 & 0 \\ 0 & 1/b & 0 \\ 0 & 0 & 1/c \end{bmatrix} \)
Reason (R) : The inverse of a diagonal matrix is a diagonal matrix.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : The rank of a unit matrix of order n × n is n.
Reason (R) : The rank of a non-singular matrix of order n × n is not n.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (c) A is true but R is false.

Question. Assertion (A) : The matrix \( \begin{bmatrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \end{bmatrix} \) is a diagonal matrix.
Reason (R) : \( A = [a_{ij}] \) is a square matrix such that \( a_{ij} = 0 \forall i \neq j \), then A is called diagonal matrix.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A) : The inverse of the matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 7 \end{bmatrix} \) does not exist.
Reason (R) : The matrix A is singular.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : If A is a matrix of order n × n, then det (kA) = \( k^n \) det (A) or | kA | = \( k^n \) | A|.
Reason (R) : If B is a matrix obtained from A by multiplying any row or column by a scalar k, then det B = k det A or | B | = k | A|.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : The matrix \( A = \frac{1}{3} \begin{bmatrix} 1 & -2 & 2 \\ -2 & 1 & 2 \\ -2 & -2 & -1 \end{bmatrix} \) is an orthogonal matrix.
Reason (R) : If A and B are orthogonal, then AB is also orthogonal.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both (A) and (R) are true but R is not correct explanation of A.

Question. Assertion (A) : If A is a skew-symmetric matrix of order 3 × 3, then det (A) = 0 or | A | = 0.
Reason (R) : If A is a square matrix, then det (A) = det (A′) = det (– A′).
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (c) A is true but R is false.

Question. Assertion (A) : The inverse of \( A = \begin{bmatrix} 3 & 4 \\ 3 & 5 \end{bmatrix} \) does not exist.
Reason (R) : The matrix A is non-singular.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A) : If a matrix of order 2 × 2, commutes with every matrix of order 2 × 2, then it is scalar matrix.
Reason (R) : A scalar matrix of order 2 × 2 commutes with every 2 × 2 matrix.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : The determinant of a matrix \( A = [a_{ij}]_{5 \times 5} \) where \( a_{ij} + a_{ji} = 0 \) for all i and j is zero.
Reason (R) : The determinant of a skew symmetric matrix of odd order is zero.
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Case Study 1

A firm produces three products \( P_1, P_2 \) and \( P_3 \) requiring the mix-up of three materials \( M_1, M_2 \) and \( M_3 \) per unit requirement of each product for each material (in units) is represented by matrix A as follows :
\( A = \begin{bmatrix} M_1 & M_2 & M_3 \\ P_1 & 2 & 3 & 1 \\ P_2 & 4 & 2 & 5 \\ P_3 & 2 & 4 & 2 \end{bmatrix} \)
The per unit cost of material \( M_1, M_2 \) and \( M_3 \) is Rs. 5, Rs. 10 and Rs. 5 respectively.

Question. If C represents the matrix showing cost of material, the matrix C will be expressed as :
(a) \( C = \begin{bmatrix} 10 \\ 5 \\ 5 \end{bmatrix} \)
(b) C = [5 10 15]
(c) \( C = \begin{bmatrix} 5 \\ 10 \\ 5 \end{bmatrix} \)
(d) \( C = \begin{bmatrix} 15 \\ 10 \\ 20 \end{bmatrix} \)
Answer: (c) \( C = \begin{bmatrix} 5 \\ 10 \\ 5 \end{bmatrix} \)

Question. Expressed matrix B in the order of 3 × 1 if the firm produces 100 units of each product :
(a) \( \begin{bmatrix} 100 \\ 100 \\ 100 \end{bmatrix} \)
(b) [100 100 100]
(c) \( \begin{bmatrix} 100 \\ 200 \\ 300 \end{bmatrix} \)
(d) \( \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} 100 \\ 100 \\ 100 \end{bmatrix} \)

Question. Find the total requirement of each material for 100 units of each product :
(a) \( \begin{bmatrix} 900 \\ 800 \\ 800 \end{bmatrix} \)
(b) \( \begin{bmatrix} 90 \\ 80 \\ 80 \end{bmatrix} \)
(c) \( \begin{bmatrix} 190 \\ 200 \\ 80 \end{bmatrix} \)
(d) \( \begin{bmatrix} 800 \\ 900 \\ 800 \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} 800 \\ 900 \\ 800 \end{bmatrix} \)

Question. Find per unit cost of production of each product as per the unit cost of material given above :
(a) \( \begin{bmatrix} 45 \\ 65 \\ 60 \end{bmatrix} \)
(b) \( \begin{bmatrix} 65 \\ 60 \\ 45 \end{bmatrix} \)
(c) \( \begin{bmatrix} 60 \\ 65 \\ 40 \end{bmatrix} \)
(d) \( \begin{bmatrix} 50 \\ 60 \\ 70 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} 45 \\ 65 \\ 60 \end{bmatrix} \)

Question. Find the total cost of production if the firm produces 200 units of each product :
(a) 28,000
(b) 30,000
(c) 34,000
(d) 32,000
Answer: (c) 34,000

Case Study 2

A concert is organised to earn the revenue of Rs. 1,80,000 which can be distributed among the needy people during the period of Covid-19. The concert hall has 4,000 seats which are divided into two sections A and B. The cost of a ticket in Section A is Rs. 50 and that of Section B is Rs. 40. All the seats are occupied.

Question. If x and y are number of seats of Section A and B respectively, then write the equation related to number of seats :
(a) x – y = 4000
(b) x + y = 400
(c) x + y = 4000
(d) 2x + y = 4000
Answer: (c) x + y = 4000

Question. What will be relation between x and y for the total revenue ?
(a) 5x + 4y = 18000
(b) 50x + 40y = 18000
(c) 5x + 4y = 1800
(d) x + y = 180
Answer: (a) 5x + 4y = 18000

Question. If equations are to be solved through the matrix method in the form of AX = B, then what will be matrix A and matrix B :
(a) \( A = \begin{bmatrix} 1 & 1 \\ 5 & 4 \end{bmatrix} , B = \begin{bmatrix} 4000 \\ 18000 \end{bmatrix} \)
(b) \( A = \begin{bmatrix} 5 & 4 \\ 1 & 1 \end{bmatrix} , B = \begin{bmatrix} 4000 \\ 18000 \end{bmatrix} \)
(c) \( A = \begin{bmatrix} 1 & 5 \\ 1 & 4 \end{bmatrix} , B = \begin{bmatrix} 18000 \\ 4000 \end{bmatrix} \)
(d) \( A = \begin{bmatrix} 1 & 4 \\ 1 & 5 \end{bmatrix} , B = \begin{bmatrix} 4000 \\ 18000 \end{bmatrix} \)
Answer: (a) \( A = \begin{bmatrix} 1 & 1 \\ 5 & 4 \end{bmatrix} , B = \begin{bmatrix} 4000 \\ 18000 \end{bmatrix} \)

Question. The value of X will be equal to \( A^{-1}B \), then \( A^{-1} \):
(a) \( \begin{bmatrix} -4 & 1 \\ -5 & 1 \end{bmatrix} \)
(b) \( \begin{bmatrix} -4 & 1 \\ 5 & -1 \end{bmatrix} \)
(c) \( \begin{bmatrix} 4 & -5 \\ -1 & 1 \end{bmatrix} \)
(d) \( \begin{bmatrix} -4 & 5 \\ 1 & -1 \end{bmatrix} \)
Answer: (b) \( \begin{bmatrix} -4 & 1 \\ 5 & -1 \end{bmatrix} \)

Question. How many seats are there in Section A and Section B of Concert Hall ?
(a) 2000, 4000
(b) 4000, 2000
(c) 2000, 2000
(d) 6000, 4000
Answer: (c) 2000, 2000

Case Study 3

A mobile shop purchases three types of mobiles namely LG, I-phone and samsung and sells them in two malls of Gurugram. The annual sales in units are given below :
Mall I: LG=20, I-phone=15, Samsung=35
Mall II: LG=10, I-phone=20, Samsung=30
If the sale prices of per unit of LG phone is Rs. 20,000, I-phone Rs. 50,000 and Samsung is Rs. 15,000 respectively and costs per unit for LG Rs. 15,000, I-phone Rs. 40,000 and Samsung Rs. 12,000 respectively.

Question. Represent the matrix Q as quantity sold :
(a) \( Q = \begin{bmatrix} 20 & 15 & 35 \\ 10 & 20 & 30 \end{bmatrix} \)
(b) \( Q = \begin{bmatrix} 200 & 150 & 350 \\ 100 & 200 & 300 \end{bmatrix} \)
(c) \( Q = \begin{bmatrix} 20 & 10 \\ 15 & 20 \\ 35 & 30 \end{bmatrix} \)
(d) \( Q = \begin{bmatrix} 200 & 100 \\ 150 & 200 \\ 350 & 300 \end{bmatrix} \)
Answer: (a) \( Q = \begin{bmatrix} 20 & 15 & 35 \\ 10 & 20 & 30 \end{bmatrix} \)

Question. Express the matrix S as total sales price of the item:
(a) \( S = \begin{bmatrix} 20,000 \\ 50,000 \\ 15,000 \end{bmatrix} \)
(b) S = [20,000 50,000 15,000]
(c) \( S = \begin{bmatrix} 50,000 \\ 20,000 \\ 15,000 \end{bmatrix} \)
(d) S = [2,000 5,000 1,500]
Answer: (a) \( S = \begin{bmatrix} 20,000 \\ 50,000 \\ 15,000 \end{bmatrix} \)

Question. Express the matrix C as the total cost price of each item :
(a) C = [15,000 40,000 12,000]
(b) \( C = \begin{bmatrix} 15,000 \\ 40,000 \\ 12,000 \end{bmatrix} \)
(c) C = [1,500 4,000 1,200]
(d) \( C = \begin{bmatrix} 12,000 \\ 15,000 \\ 40,000 \end{bmatrix} \)
Answer: (b) \( C = \begin{bmatrix} 15,000 \\ 40,000 \\ 12,000 \end{bmatrix} \)

Question. Express the matrix P as profit earned on each item :
(a) P = [3,000 10,000 5,000]
(b) P = [5,000 10,000 3,000]
(c) \( P = \begin{bmatrix} 5,000 \\ 10,000 \\ 3,000 \end{bmatrix} \)
(d) \( P = \begin{bmatrix} 3,000 \\ 5,000 \\ 10,000 \end{bmatrix} \)
Answer: (c) \( P = \begin{bmatrix} 5,000 \\ 10,000 \\ 3,000 \end{bmatrix} \)

Question. What is the total amount of profit in Mall I and Mall II if all goods are sold ?
(a) \( \begin{bmatrix} 35,500 \\ 34,000 \end{bmatrix} \)
(b) \( \begin{bmatrix} 3,40,000 \\ 3,55,000 \end{bmatrix} \)
(c) [34,000 35,500]
(d) \( \begin{bmatrix} 3,55,000 \\ 3,40,000 \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} 3,55,000 \\ 3,40,000 \end{bmatrix} \)

Case Study 4

Two trust A and B receive Rs. 70,000 and Rs. 55,000 respectively from central government to award prize to persons of a district in three fields defence, health services and education. Trust A awarded 10, 5 and 15 persons in the field of defence, health services and education respectively, while B awarded 15, 10 and 5 persons respectively. All three prizes amount of Rs. 6000.

Question. If x, y, z represents the amount of individual prize then write the linear equations :
(a) x + y + z = 6,000
10x + 5y + 15z = 70,000
15x + 10y + 5z = 55,000
(b) x + 10y + 15z = 6,000
x + 5y + 10z = 70,000
x + 15y + 5z = 55,000
(c) x – y + z = 6,000
5y + 15z = 70,000
15x + 5z = 55,000
(d) x – y – z = 6,000
x – 5y + 10z = 70,000
15x + 10y + 5z = 55,000
Answer: (a) x + y + z = 6,000; 10x + 5y + 15z = 70,000; 15x + 10y + 5z = 55,000

Question. If all the equations are written in the form of AX = B then find |A| :
(a) 100
(b) 175
(c) 75
(d) 200
Answer: (c) 75

Question. Find adj A :
(a) \( \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & -5 \end{bmatrix} \)
(b) \( \begin{bmatrix} -125 & 175 & 25 \\ 5 & -10 & 5 \\ 10 & -5 & -5 \end{bmatrix} \)
(c) \( \begin{bmatrix} 175 & -10 & -5 \\ 25 & 5 & -5 \\ -125 & 5 & 10 \end{bmatrix} \)
(d) \( \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & 5 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & -5 \end{bmatrix} \)

Question. Find \( A^{-1} \) :
(a) \( \frac{1}{75} \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & -5 \end{bmatrix} \)
(b) \( \frac{1}{75} \begin{bmatrix} -125 & 175 & 25 \\ 5 & -10 & 5 \\ 10 & -5 & -5 \end{bmatrix} \)
(c) \( \frac{1}{75} \begin{bmatrix} 175 & -10 & -5 \\ 25 & 5 & -5 \\ -125 & 5 & 10 \end{bmatrix} \)
(d) \( \frac{1}{75} \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & 5 \end{bmatrix} \)
Answer: (a) \( \frac{1}{75} \begin{bmatrix} -125 & 5 & 10 \\ 175 & -10 & -5 \\ 25 & 5 & -5 \end{bmatrix} \)

Question. Find the value of each prize :
(a) Rs. 2000, Rs. 3000, Rs. 1000
(b) Rs. 2000, Rs. 1000, Rs. 3000
(c) Rs. 1000, Rs. 2000, Rs. 3000
(d) Rs. 3000, Rs. 2000, Rs. 1000
Answer: (b) Rs. 2000, Rs. 1000, Rs. 3000

Case Study 5

Three shopkeepers Ram Lal, Shyam Lal and Ghansham are using polythene bags handmade bags (prepared by prisoners), and newspaper’s envelope as carry bags. It is found that the shopkeepers Ram Lal Shyam Lal and Ghansham are using (20, 30, 40), (30, 40, 20) and (40, 20, 30) polythene bags handmade bags and newspapers envelopes respectively. The shopkeepers Ram Lal, Shyam Lal and Ghansham spent Rs. 250, Rs. 270 and Rs. 200 on these carry bags respectively.

Question. What is the cost of one polythene bag?
(a) Rs. 1
(b) Rs. 2
(c) Rs. 3
(d) Rs. 5
Answer: (a) Rs. 1

Question. What is the cost of one handmade bag?
(a) Rs. 1
(b) Rs. 2
(c) Rs. 3
(d) Rs. 5
Answer: (d) Rs. 5

Question. What is the cost of one newspaper bag?
(a) Rs. 1
(b) Rs. 2
(c) Rs. 3
(d) Rs. 5
Answer: (b) Rs. 2

Question. Keeping in mind the social conditions, which shopkeeper is better?
(a) Ram Lal
(b) Shyam Lal
(c) Ghansham
(d) None of the options
Answer: (b) Shyam Lal

Question. Keeping in mind the environmental conditions, which shopkeeper is better?
(a) Ram Lal
(b) Shyam Lal
(c) Ghansham
(d) None of the options
Answer: (a) Ram Lal