Question 1:
In the matrix , write:
(i) The order of the matrix (ii) The number of elements,
(iii) Write the elements a_{13}, a_{21}, a_{33}, a_{24}, a_{23}
Answer
(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 × 4.
(ii) Since the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.
(iii) a_{13} = 19, a_{21} = 35, a_{33} = −5, a_{24} = 12, a_{23} =
If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
Answer
We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.
The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and
(6, 4)
Hence, the possible orders of a matrix having 24 elements are: 1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4
(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13. Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.
If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer
We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18.
The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3) Hence, the possible orders of a matrix having 18 elements are:
1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3
(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5. Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.
Construct a 3 × 4 matrix, whose elements are given by
(i) (ii)
Answer
In general, a 3 × 4 matrix is given by
Therefore, the required matrix is
Therefore, the required matrix is
Find the value of x, y, and z from the following equation:
(i) (ii)
(iii)
Answer
As the given matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
x = 1, y = 4, and z = 3
As the given matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
x + y = 6, xy = 8, 5 + z = 5 Now, 5 + z = 5 ⇒ z = 0
We know that:
(x − y)^{2} = (x + y)^{2} − 4xy
⇒ (x − y)^{2} = 36 − 32 = 4
⇒ x − y = ±2
Now, when x − y = 2 and x + y = 6, we get x = 4 and y = 2 When x − y = − 2 and x + y = 6, we get x = 2 and y = 4
∴x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0
As the two matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
x + y + z = 9 … (1)
x + z = 5 … (2)
y + z = 7 … (3)
From (1) and (2), we have:
y + 5 = 9
⇒ y = 4
Then, from (3), we have: 4 + z = 7
⇒ z = 3
∴ x + z = 5
⇒ x = 2
∴ x = 2, y = 4, and z = 3
Find the value of a, b, c, and d from the equation:
Answer
As the two matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
a − b = −1 … (1) 2a − b = 0 … (2)
2a + c = 5 … (3)
3c + d = 13 … (4)
From (2), we have:
b = 2a
Then, from (1), we have:
a − 2a = −1
⇒ a = 1
⇒ b = 2
Now, from (3), we have: 2 ×1 + c = 5
⇒ c = 3
From (4) we have:
3 ×3 + d = 13
⇒ 9 + d = 13 ⇒ d = 4
∴a = 1, b = 2, c = 3, and d = 4
is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these Answer
The correct answer is C.
It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns.
Therefore, is a square matrix, if m = n.
Which of the given values of x and y make the following pair of matrices equal
(B) Not possible to find
(D)
Answer
The correct answer is B.
We find that on comparing the corresponding elements of the two matrices, we get two different values of x, which is not possible.
Hence, it is not possible to find the values of x and y for which the given matrices are equal.
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 Answer
The correct answer is D.
The given matrix of the order 3 × 3 has 9 elements and each of these elements can be either 0 or 1.
Now, each of the 9 elements can be filled in two possible ways.
Therefore, by the multiplication principle, the required number of possible matrices is 2^{9}
= 512
Question 1:
Let Find each of the following
(i) (ii) (iii)
(iv) (v)
Answer
(ii)
(iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B. Therefore, AB is defined as:
(v)
Compute the following:
(i) (ii)
(iii)
(v)
Answer
(ii)
(iv)
Compute the indicated products (i)
(ii)
(iii)
(iv)
(v)
(vi)
Answer
(ii)
(iv)
(vi)
If , and , then
compute and . Also, verify that Answer
If and then compute .
Answer
Find X and Y, if
(i) and
(ii) and
Answer
Multiplying equation (3) with (2), we get:
Multiplying equation (4) with (3), we get:
From (5) and (6), we have:
Now,
Find X, if and
Answer
Comparing the corresponding elements of these two matrices, we have:
∴x = 3 and y = 3
Solve the equation for x, y, z and t if
Answer
Comparing the corresponding elements of these two matrices, we get:
If , find values of x and y.
Answer
Comparing the corresponding elements of these two matrices, we get: 2x − y = 10 and 3x + y = 5
Adding these two equations, we have:
5x = 15
⇒ x = 3
Now, 3x + y = 5
⇒ y = 5 − 3x
⇒ y = 5 − 9 = −4
∴x = 3 and y = −4
Given , find the values of x, y, z and
Comparing the corresponding elements of these two matrices, we get:
If , show that . Answer
Show that
(i)
(ii)
Answer
(ii)
Find if
Answer
We have A^{2} = A × A
If , prove that
Comparing the corresponding elements, we have:
Thus, the value of k is 1.
If and I is the identity matrix of order 2, show that
Answer
A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
(a) Rs 1,800 (b) Rs 2,000
Answer
(a) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).
It is given that the first bond pays 5% interest per year and the second bond pays 7% interest per year.
Therefore, in order to obtain an annual total interest of Rs 1800, we have:
Thus, in order to obtain an annual total interest of Rs 1800, the trust fund should invest Rs 15000 in the first bond and the remaining Rs 15000 in the second bond.
(b) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).
Therefore, in order to obtain an annual total interest of Rs 2000, we have:
Thus, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.
The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Answer
The bookshop has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen economics books.
The selling prices of a chemistry book, a physics book, and an economics book are respectively given as Rs 80, Rs 60, and Rs 40.
Thus, the bookshop will receive Rs 20160 from the sale of all these books.
Assume X, Y, Z, W and P are matrices of order , and respectively. The restriction on n, k and p so that will be defined are:
A. k = 3, p = n
B. k is arbitrary, p = 2
C. p is arbitrary, k = 3
D. k = 2, p = 3
Answer
Matrices P and Y are of the orders p × k and 3 × k respectively.
Therefore, matrix PY will be defined if k = 3. Consequently, PY will be of the order p × k. Matrices W and Y are of the orders n × 3 and 3 × k respectively.
Since the number of columns in W is equal to the number of rows in Y, matrix WY is well-defined and is of the order n × k.
Matrices PY and WY can be added only when their orders are the same.
However, PY is of the order p × k and WY is of the order n × k. Therefore, we must have
p = n.
Thus, k = 3 and p = n are the restrictions on n, k, and p so that will be defined.
Assume X, Y, Z, W and P are matrices of order , and respectively. If n = p, then the order of the matrix is
A p × 2 B 2 × n C n × 3 D p × n
Answer
The correct answer is B. Matrix X is of the order 2 × n.
Therefore, matrix 7X is also of the same order.
Matrix Z is of the order 2 × p, i.e., 2 × n [Since n = p] Therefore, matrix 5Z is also of the same order.
Now, both the matrices 7X and 5Z are of the order 2 × n. Thus, matrix 7X − 5Z is well-defined and is of the order 2 × n.
Question 1:
Find the transpose of each of the following matrices:
(i) (ii) (iii)
Answer
(ii)
Question 2:
If and , then verify that
(i)
(ii) Answer
We have:
(ii)
If and , then verify that (i)
(ii)
Answer
(i)
Question 4:
If and , then find Answer
We know that
For the matrices A and B, verify that (AB)′ = where
(i)
(ii)
Answer
(ii)
If (i) , then verify that
(ii) , then verify that
Answer
(ii)
(i) Show that the matrix is a symmetric matrix
(ii) Show that the matrix is a skew symmetric matrix
Answer
(i) We have:
Hence, A is a symmetric matrix.
(ii) We have:
Hence, A is a skew-symmetric matrix.
For the matrix , verify that
(i) is a symmetric matrix
(ii) is a skew symmetric matrix Answer
Hence, is a symmetric matrix.
Hence, is a skew-symmetric matrix.
Find and , when
Answer
The given matrix is
Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i)
(ii)
(iii)
(iv)
Answer
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix. Representing A as the sum of P and Q:
If A, B are symmetric matrices of same order, then AB − BA is a
A. Skew symmetric matrix B. Symmetric matrix
C. Zero matrix D. Identity matrix Answer
The correct answer is A.
A and B are symmetric matrices, therefore, we have:
Thus, (AB − BA) is a skew-symmetric matrix.
If , then , if the value of α is
C. π D.
Answer
The correct answer is B.
Comparing the corresponding elements of the two matrices, we have:
Exercise 3.4
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Now, in the above equation, we can see all the zeros in the second row of the matrix on the L.H.S.
Therefore, A^{−1} does not exist.
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Applying , we have:
Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S.
Therefore, A^{−1} does not exist.
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Applying R_{2} → R_{2} + 3R_{1} and R_{3} → R_{3} − 2R_{1}, we have:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Applying , we have:
Matrices A and B will be inverse of each other only if
A. AB = BA
C. AB = 0, BA = I
B. AB = BA = 0
D. AB = BA = I
Answer
We know that if A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is said to be the inverse of
A. In this case, it is clear that A is the inverse of B.
Thus, matrices A and B will be inverses of each other only if AB = BA = I.
Question 1:
Let , show that , where I is the identity matrix of order 2 and n ∈ N
Answer
It is given that
We shall prove the result by using the principle of mathematical induction. For n = 1, we have:
Therefore, the result is true for n = 1. Let the result be true for n = k.
That is,
Now, we prove that the result is true for n = k + 1. Consider
From (1), we have:
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:
If , prove that
Answer
It is given that
We shall prove the result by using the principle of mathematical induction. For n = 1, we have:
Therefore, the result is true for n = 1. Let the result be true for n = k.
That is
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus by the principle of mathematical induction, we have:
If , then prove where n is any positive integer Answer
It is given that
We shall prove the result by using the principle of mathematical induction. For n = 1, we have:
Therefore, the result is true for n = 1. Let the result be true for n = k.
That is,
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix. Answer
It is given that A and B are symmetric matrices. Therefore, we have:
Thus, (AB − BA) is a skew-symmetric matrix.
Show that the matrix is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer
Thus, if A is a symmetric matrix, then is a symmetric matrix. Now, we suppose that A is a skew-symmetric matrix.
Then,
Thus, if A is a skew-symmetric matrix, then is a skew-symmetric matrix.
Hence, if A is a symmetric or skew-symmetric matrix, then is a symmetric or skew- symmetric matrix accordingly.
Solve system of linear equations, using matrix method.
Answer
The given system of equations can be written in the form of AX = B, where
Thus, A is non-singular. Therefore, its inverse exists.
For what values of ?
∴4 + 4x = 0
⇒ x = −1
Thus, the required value of x is −1.
If , show that
Answer
Find x, if
Answer We have:
A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:
Market |
Products |
||
I |
10000 |
2000 |
18000 |
II |
6000 |
20000 |
8000 |
(a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.
Answer
(a) The unit sale prices of x, y, and z are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00.
The total revenue in market II can be represented in the form of a matrix as:
Therefore, the total revenue in market I isRs 46000 and the same in market II isRs 53000.
(b) The unit cost prices of x, y, and z are respectively given as Rs 2.00, Rs 1.00, and 50 paise.
Since the total revenue in market I isRs 46000, the gross profit in this marketis (Rs 46000 − Rs 31000) Rs 15000.
Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs 53000 − Rs 36000) Rs 17000.
Find the matrix X so that Answer
It is given that:
The matrix given on the R.H.S. of the equation is a 2 × 3 matrix and the one given on the L.H.S. of the equation is a 2 × 3 matrix. Therefore, X has to be a 2 × 2 matrix.
Equating the corresponding elements of the two matrices, we have:
Thus, a = 1, b = 2, c = −2, d = 0
Hence, the required matrix X is
If A and B are square matrices of the same order such that AB = BA, then prove by
induction that . Further, prove that for all n ∈ N
Answer
A and B are square matrices of the same order such that AB = BA.
For n = 1, we have:
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have Now, we prove that for all n ∈ N
For n = 1, we have:
Therefore, the result is true for n = 1. Let the result be true for n = k.
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have , for all natural numbers.
Choose the correct answer in the following questions:
If is such that then
B.
D.
Answer
On comparing the corresponding elements, we have:
If the matrix A is both symmetric and skew symmetric, then
A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these Answer Answer: B
If A is both symmetric and skew-symmetric matrix, then we should have
Therefore, A is a zero matrix.
If A is square matrix such that then is equal to
A. A B. I − A C. I D. 3A Answer