matrix , chapter 3 mathematics class 12

S.No.	Matrix chapter 3
	EXERCISE 3.1
1	In the matrix A, write (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23.
2	If a matrix has 24 elements, what are the possible orders it can have? What, if ithas 13 elements?
3	If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
4	Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
5	Construct a 3 × 4 matrix, whose elements are given by:
6	Find the values of x, y and z from the following equations:
7	Find the value of a, b, c and d from the equation:
8	A = [aij]m × n\ is a square matrix, if (A) m < n (B) m > n (C) m = n (D) None of these
9	Which of the given values of x and y make the following pair of matrices equal
10	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
	EXERCISE 3.2
1	Let A= , B= , and C= Find each of the following: (i) A + B (ii) A – B (iii) 3A – C (iv) AB (v) BA
2	Compute the following:
3	Compute the indicated products.
4	If A= , B= , and C= then compute (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.
5	If A = , B = , then compute the 3A- 5B
6	Simplify
7	Find X and Y, if
8	Find X, if Y = and and 2X + Y =
9	Find x and y, if
10	Solve the equation for x, y, z and t, if
11	if find the values of x and y.
12	Given , find the values of x, y, z and w.
13	If F(x) = show that F(x) F(y) = F(x + y).
14	Show tha
15	Find A2 – 5A + 6I, if A=
16	If A = prove that A3 – 6A2 + 7A + 2I = 0
17	If A = , k = find k so that A2 = kA – 2I
18	If A = and I is the identity matrix of order 2, show that
19	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 1800 (b) Rs 2000

20	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.
21	Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Choose the correct answer in Exercises 21 and 22.
22	21. The restriction on n, k and p so that PY + WY 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
23	If n = p, then the order of the matrix 7X – 5Z is: (A) p × 2 (B) 2 × n (C) n × 3 (D) p × n
	EXERCISE 3.3
1	Find the transpose of each of the following matrices:
2	If A = and B = , then verify that (i) (A + B)′ = A′ + B′, (ii) (A – B)′ = A′ – B′
3	If A′ = and B = ,then verify that (i) (A + B)′ = A′ + B′ (ii) (A – B)′ = A′ – B′
4	If A′ = and B = ,then find (A + 2B)′
5	For the matrices A and B, verify that (AB)′ = B′A′, where
6	If A = then verify that A′ A = I
7	(i) Show that the matrix A= is a symmetric matrix. (ii) Show that the matrix A = is a skew symmetric matrix
8	For the matrix A= , verify that (i) (A + A′) is a symmetric matrix (ii) (A – A′) is a skew symmetric matrix
9	Find when A =
10	Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
	EXERCISE 3.4
1	Using elementary transformations, find the inverse of each of the matrices, if it exists in Exercises 1 to 17.
1	Question 1
2	Question 2
3	Question 3
4	Question 4
5	Question 5
6	Question 6
7	Question 7
8	Question 8
9	Question 9
10	Question 10
11	Question 11
12	Question 12
13	Question 13
14	Question 14
15	Question 15
16	Question 16
17	Question 17
18	Matrices A and B will be inverse of each other only if (A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I
	Miscellaneous Exercise on Chapter 3
1	Let A=, show that (aI + bA)n = an I + nan – 1 bA, where I is the identity matrix of order 2 and n ∈ N.
2	If A = , prove that n ∈N
3	If A = , then prove that An = , where n is any positive integer.
4	If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
5	Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
6	Find the values of x, y, z if the matrix A= satisfy the equation A′A = I.
7	For what values of x :
8	If A = , show that A2 – 5A + 7I = 0.
9	Find x, if
10	A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000 2,000 18,000 II 6,000 20,000 8,000 (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.
11	Find the matrix X so that
12	If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that (AB)n = AnBn for all n ∈ N.
	Choose the correct answer in the following questions:
13	If A =is such that A² = I, then(A) 1 + α² + βγ = 0 (B) 1 – α² + βγ = 0 (C) 1 – α² – βγ = 0 (D) 1 + α² – βγ = 0
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
15	If A is square matrix such that A2 = A, then (I + A)³ – 7 A is equal to (A) A (B) I – A (C) I (D) 3A