RD SHARMA Solutions for Class 12science Maths Chapter 5  Algebra of Matrices
Chapter 5  Algebra of Matrices Exercise MCQ
If A and B are symmetric matrices of the same order, then AB^{T}  BA^{T} is a
a. skewsymmetric matrix
b. null matrix
c. symmetric matrix
d. none of these
As A and B are symmetric matrices, we have
A^{T} = A and B^{T} = B … (i)
Consider,
Hence, AB^{T}  BA^{T} is a skewsymmetric matrix.
a. a null matrix
b. a unit matrix
c. A
d. A
a.
b.
c.
d.
If A and B are two matrices such that AB = A and BA = B, then B^{2} is equal to
a. B
b. A
c. 1
d. 0
If AB = A and BA = B, where A and B are square matrices, then
a. B^{2} = B and A^{2} = A
b. B^{2}≠ B and A^{2} = A
c. A^{2}≠ A,B^{2} = B
d. A^{2}≠ A, B^{2}≠ B
If A and B are two matrices such that AB = B and BA =A, then A^{2} + B^{2} is equal to
a. 2 AB
b. 2 BA
c. A + B
d. AB
a. 3
b. 4
c. 6
d. 7
If the matrix AB is zero , then
a. It is not necessary that either A = 0 or B = 0
b. A = 0 or B = 0
c. A = O and B = 0
d. All the above statements are wrong
a.
b.
c.
d.
If A, B are square matrices of order 3, A is nonsingular and AB = 0, then B is a
a. Null matrix
b. Singular matrix
c. Unit matrix
d. Nonsingular matrix
a. B
b. nB
c. B^{n}
d. A + B
a.
b.
c.
d.
a. 0
b. 1
c. 2
d. None of these
a. a = 4, b = 1
b. a = 1, b = 4
c. a = 0, b = 4
d. a = 2, b = 4
a. 1 + α^{2} + βγ = 0
b. 1  α^{2} + βγ = 0
c. 1  α^{2}  βγ = 0
d. 1 + α^{2}  βγ = 0
If S = [s_{ij}] is a scalar matrix such that s_{ii} = k and A is a square matrix of the same order, then AS = SA = ?
a. A^{k}
b. k + A
c. kA
d. kS
If A is a square matrix such that A^{2} = A, then (I + A)^{3}  7A is equal to
a. A
b. I  A
c. I
d. 3A
If a matrix A is both symmetric and skewsymmetric, then
a. A is a diagonal matrix
b. A is a zero matrix
c. A is a scalar matrix
d. A is a square matrix
a. A skewsymmetric matrix
b. A symmetric matrix
c. A diagonal matrix
d. An upper triangular matrix
If A is a square matrix, then AA is a
a. Skewsymmetric matrix
b. Symmetric matrix
c. Diagonal matrix
d. None of these
If A and B are symmetric matrices, then ABA is
a. Symmetric matrix
b. Skewsymmetric matrix
c. Diagonal matrix
d. Scalar matrix
a. X = 0, y = 0
b. X + y = 5
c. X = y
d. None of these
If A is 3 × 4 matrix and B is a matrix such that A^{T}B and BA^{T} are both defined. Then, B is of the type
a. 3 × 4
b. 3 × 3
c. 4 × 4
d. 4 × 3
If A = [a_{ij}] is a square matrix of even order such that a_{ij} = i^{2}  j^{2}, then
a. A is a skew  symmetric matrix and A = 0
b. A is symmetric matrix and A is a square
c. A is symmetric matrix and A = 0
d. None of these
a.
b.
c.
d. none of these
a.
b.
c.
d.
Out of the following matrices, choose that matrix which is a scalar matrix:
a.
b.
c.
d.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
a. 27
b. 18
c. 81
d. 512
Which of the given values of x and y make the following pairs of matrices equal?
a.
b.
c.
d. Not possible to find
a. 6, 12, 18
b. 6, 4, 9
c. 6, 4, 9
d. 6, 12, 18
a. I cos θ + J sin θ
b. I sin θ + J cos θ
c. I cos θ  J sin θ
d.  I cos θ + J sin θ
a. 17
b. 25
c. 3
d. 12
If A = [a_{ij}] is a scalar matrix of order n × n such that a_{ii} = k for all I, then trace of A is equal to
a. nk
b. n + k
c.
d. none of these
a. square matrix
b. diagonal matrix
c. unit matrix
d. none of these
The number of possible matrices of order 3 × 3 with each entry 2 or 0 is
a. 9
b. 27
c. 81
d. none of these
a. x = 3, y = 1
b. x = 2, y = 3
c. x = 2, y = 3
d. x = 3, y = 3
If A is a square matrix such that A^{2} = I, then (A  I)^{3} + (A + I)^{3}  7A is equal to
a. A
b. I  A
c. I + A
d. 3A
If A and B are two matrix of order 3 × m and 3 × n respectively and m = n, then the order of 5A  2B is
a. m × n
b. 3 × 3
c. m × n
d. 3 × n
If A is a matrix of order m × n and B is a matrix such that AB^{T} and B^{T}A are both defined, then the order of matrix B is
a. m × n
b. n × n
c. n × m
d. 3 × n
If A and B are matrices of the same order, then (AB^{T}BA^{T})^{T} is a
a. skewsymmetric matrix
b. null matrix
c. unit matrix
d. symmetric matrix
a. I
b. A
c. O
d. I
a. I
b. 0
c. 2I
d.
If A and B are square matrices of the same order, then (A + B) (A  B) is equal to
a. A^{2}  B^{2}
b. A^{2}  BA  AB  B^{2}
c. A^{2}  B^{2} + BA  AB
d. A^{2}  BA + B^{2} + AB
a. Only AB is defined
b. Only BA is defined
c. AB ad BA both are defined
d. AB and BA both are not defined
a. Diagonal matrix
b. Symmetric matrix
c. Skewsymmetric matrix
d. Scalar matrix
a. Identity matrix
b. Symmetric matrix
c. Skewsymmetric matrix
d. Diagonal matrix
Correct option: (d)
A matrix is called Diagonal matrix if all the elements, except those in the leading diagonal, are zero.
Chapter 5  Algebra of Matrices Exercise Ex. 5.1
If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?
We know that if a matrix is of the order , it has mn elements. Thus, to find all the possible orders of a matrix having 8 elements, we have to find all the ordered pairs of natural numbers whose products is 8.
The ordered pairs are:
are the ordered pairs of natural numbers whose product is 5.
Hence, the possible orders of a matrix having 5 elements are
Construct a 2 × 2 matrix A = [a_{ij}] whose elements aij are given by:
Construct a 2 × 2 matrix A = [a_{ij}] whose elements a_{ij} are given by:
A_{ij} = e^{2ix} sin xj
Construct a 3 × 4 matrix A = [a_{ij}] whose elements aij are given by :
(i) aij = i + j
(ii) aij = i  j
(iii) aij = 2i
(iv) aij = j
The sales figure of two car dealer during january 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of january  february revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. in the same 2 month period, dealer b sold 10 deluxe, 5 premium and 7 standard cars. Write 2 × 3 matrices summarizing for january and 2  month period for each dealer.
Find the values of a and b if A = B, where
As A = B,
Chapter 5  Algebra of Matrices Exercise Ex. 5.2
(ii)
Find x, y satisfying the matrix equations
(i)
(ii)
(i)
(ii)
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs. 15000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Let 3x and 4x be the monthly incomes of Aryan and Babban respectively.
Let 5y and 7y be their monthly expenditures respectively.
As each individual saves Rs. 15000 per month, we have
3x  5y = 15000 … (i)
4x  7y = 15000 … (ii)
The above equations can be written in matrix form as follows
Let AX = B, where
Let's find A^{1}
Therefore, x = Rs. 30000 and y = Rs. 15000
So, monthly income of Aryan = Rs. 90,000 and monthly income of Babban is Rs. 120,000.
Chapter 5  Algebra of Matrices Exercise Ex. 5.3
Use this to find A^{4}
Find the matrix A such that
Find the matrix A such that
Then show that (A + B)^{2} = A^{2} + B^{2}.
Let A and B be square matrices of the order 3 × 3.
Is (AB)^{2} = A^{2} B^{2}? Give reasons.
If A and B be square matrices of the same order such that AB = BA, then show that (A + B)^{2} = A^{2} + 2AB + B^{2}.
To promote making of toilets for women, an organization tried to generate awareness through (i) house calls (ii) letters and (iii) announcements. The cost for each mode per attempt is given below:
i. Rs. 50
ii. Rs. 20
iii. Rs. 40
The number of attempts made in three villages X, Y, and Z are given below:

(i) 
(ii) 
(iii) 
X 
400 
300 
100 
Y 
300 
250 
75 
z 
500 
400 
150 
Find the total cost incurred by the organization for three villages separately, using matrices.
There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, Calculate the total requirement of calories and proteins for each of the families. What awareness can you create among people about the planned diet from this question?
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways  telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
The number of contacts of each type made in two cities X and Y is given in the matrix B as
Find the total amount spent by the party in the two cities.
What should one consider before casting his/her vote  party's promotional activity or their social activities?
If A is a square matrix such that A^{2} = A, then find (2 + A)^{3}  19A.
Given: A^{2} = A
(2 + A)^{3}  19A
= 8I + A^{3} + 12A + 6A^{2}  19A
= A^{3} + 6A^{2}  7A + 8I
= A^{2} + 6A  7A + 8I
= A  A + 8I
= 8I
Hence, (2 + A)^{3}  19A = 8I.
Solve the matrix equation:
Find the matrix A such that
Given:
Let
Hence,
Find the matrix A such that
Given:
Let
Solving equations (i) and (ii), we get, a_{1} = 1, b_{1} = 2
Solving (iii) and (iv), we get, a_{2} = 2, b_{2} = 0
From equations (v) and (vi), we get, a_{3} = 5, b_{3} = 4
Hence,
If and find scalar k so that A^{2} + I = kA.
Given: and
As A^{2} + I = kA
Hence, k = 4.
If A is a square matrix, using mathematical induction prove that for all n ∈ N.
To prove
For n = 1,
Therefore, it is true for n = 1.
Suppose the result is true for n = k
Take n = k + 1
Thus, is true for all n ∈ N.
The monthly incomes of Aryan and Babbar are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs. 15000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Let 3x and 4x be the monthly incomes of Aryan and Babbar respectively.
Let 5y and 7y be their monthly expenditures respectively.
As each individual saves Rs. 15000 per month, we have
3x  5y = 15000 … (i)
4x  7y = 15000 … (ii)
The above equations can be written in matrix form as follows
Let AX = B, where
Let's find A^{1}
Therefore, x = Rs. 30000 and y = Rs. 15000
So, monthly income of Aryan = Rs. 90,000 and monthly income of Babbar is Rs. 120,000.
This encourages us to understand the power of savings and we should save certain money for future.
A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs. 2800 as interest. However, if trust had interchanged money in bonds, they would have got Rs. 100 less as interest. Using matrix method, find the amount invested by the trust.
Let Rs. x and Rs. y is being invested in the first and second bonds respectively.
Let A be the investment matrix and B be the interest matrix.
Therefore,
The annual interest = AB =
If the interest had been interchanged, the total interest would be Rs. 100 less.
Equations (i) and (ii) can be expressed as
PX = Q, where
Now, P = 100  144 = 44
So, inverse of P exist.
Thus, x = 10000 and y = 15000
Hence, the total amount invested is Rs. 25,000.
Chapter 5  Algebra of Matrices Exercise Ex. 5.4
If l_{i}, m_{i}, n_{i} ; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^{T} = I,
Chapter 5  Algebra of Matrices Exercise Ex. 5.5
For the matrix find A + A^{T} and verify that it is a symmetric matrix.
Given:
Consider,
Thus, A + A^{T} is a symmetric matrix.
If is a symmetric matrix, then find the value of x.
As is a symmetric matrix, its transpose will be equal to itself.
Hence, the value of x is 5.
Chapter 5  Algebra of Matrices Exercise Ex. 5VSAQ
Matrix is given to be symmetric, find the values of a and b.
As A is a symmetric matrix, we have
A^{T} = A
Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.
Number of elements in a 2 × 2 matrix = 4
The first element can be 1, 2 or 3.
The second element can be 1, 2 or 3.
Similarly, the remaining two elements can take either of the 3 numbers.
So, for every element we have 3 choices.
Therefore, number of ways of writing 1, 2 or 3 in a 2 × 2 matrix is 3^{4} which is 81.
Thus, the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3 is 81.
If then write the order of matrix A.
Let
Order of matrix P is 1 × 3
Order of matrix Q is 3 × 3
Order of matrix R is 3 × 1
After multiplying P and Q, we'll get an output matrix B of order 1 × 3.
After multiplying B with R, we'll get an output matrix of order 1 × 1.
Hence, the order of matrix A is 1 × 1.
If is written as A = P + Q, whereas A = P + Q, where P is symmetric and Q is skewsymmetric matrix, then write the matrix P.
As P is symmetric, we have
As Q is skewsymmetric, we have
Let A and B be matrices of orders 3 × 2 and 2 × 4 respectively. Write the order of matrix AB.
Matrix A is order 3 × 2
Matrix B is of order 2 × 4
Then the product matrix AB will have the order 3 × 4.
If the matrix is skewsymmetric, find the values of 'a' and 'b'.
As matrix A is skewsymmetric
Therefore, A = A^{T}
Hence, the values of a and b are 2 and 3 respectively.
If A = is written as B + C, where B is a symmetric matrix and C is a skew symmetric matrix, then find B.
If a matrix has 5 elements, write all possible orders it can have.
If a matrix is of order , then the number of elements in the matrix is the product .
Given that the required matrix is having 5 elements and 5 is a prime number.
Hence the prime factorization of 5 is either .
Thus, the order of the matrix is either .
If A is a square matrix such that A^{2}=A, then write the value of 7 A(I+A)^{3}, where I is the identity matrix.
A^{2} = A
A^{3} = A^{2} = A
7A  (I + A)^{3}
= 7A  (I^{3} + A^{3} + 3A^{2}I + 3AI^{2})
= 7A  (I + A + 3A + 3A)
= 7A  (I + 7A)
= I
Write 2×2 matrix which is both symmetric and skewsymmetric.
Construct a 2 × 2 matrix A = [aij] whose elements aij are given by
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