Matrices
Miscellaneous Solutions
Question 1:
Let where I is the identity
matrix of order 2 and n ε N
Answer
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:
Question 2:
Answer
Question 3:
Answer
Question 4:
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
Answer
Thus, (AB − BA) is a skewsymmetric matrix.
Question 5:
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric
or skew symmetric.
Answer
We suppose that A is a symmetric matrix, then A' = A… (1)
Consider
Thus, if A is a skewsymmetric matrix, then B'AB is a skewsymmetric matrix.
Hence, if A is a symmetric or skewsymmetric matrix, then B'AB is a symmetric or
skewsymmetric matrix accordingly.
Question 6:
Solve system of linear equations, using matrix method.
Answer
Question 7:
For what values of
Answer
Question 8:
If
Answer
Question 9:
Find x, if
Answer
We have:
Question 10:
A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
(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:
Question 12:
If A and B are square matrices of the same order such that AB = BA, then prove by
induction that for all n ε N
Answer
A and B are square matrices of the same order such that AB = BA.
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:
Answer: C
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 skewsymmetric matrix, then we should have
Therefore, A is a zero matrix.
Question 15:
Answer
Answer: C
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