EXERCISE 3.5
Page No 100:
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:
Question 2:
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:
Question 3:
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:
Question 4:
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.
Question 5:
Show that the matrix is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:
We suppose that A is a symmetric matrix, then … (1)
Consider
Thus, if A is a symmetric matrix, thenis a symmetric matrix.
Now, we suppose that A is a skew-symmetric matrix.
Then,
Thus, if A is a skew-symmetric matrix, thenis a skew-symmetric matrix.
Hence, if A is a symmetric or skew-symmetric matrix, thenis a symmetric or skew-symmetric matrix accordingly.
Question 6:
Find the values of x, y, z if the matrix satisfy the equation
Answer:
Now,
On comparing the corresponding elements, we have:
Question 7:
For what values of?
Answer:
We have:
∴4 + 4x = 0
⇒ x = −1
Thus, the required value of x is −1.
Question 8:
If, show that
Answer:
Question 9:
Find x, if
Answer:
We have:
Page No 101:
Question 10:
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.
Consequently, the total revenue in market I can be represented in the form of a matrix as:
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.
Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:
Since the total revenue in market I isRs 46000, the gross profit in this marketis (Rs 46000 − Rs 31000) Rs 15000.
The total cost prices of all the products in market II can be represented in the form of a matrix as:
Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs 53000 − Rs 36000) Rs 17000.
Question 11:
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.
Now, let
Therefore, we have:
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
Question 12:
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:
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
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.
Question 13:
Choose the correct answer in the following questions:
If is such that then
A.
B.
C.
D.
Answer:
Answer: C
On comparing the corresponding elements, we have:
Question 14:
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.
Question 15:
If A is square matrix such that then is equal to
A. A B. I − A C. I D. 3A
Answer:
Answer: C