HOLIDAY ASSIGNMENT SUBJECT : MATHEMATICS

𝑖
1. For a 2 × 2 matrix, 𝐴 = [𝑎𝑖𝑗 ] whose elements are given by 𝑎𝑖𝑗 = 𝑗 , write the value of 𝑎12
1
2. Write the order of the product matrix 2 [2 3 4]
3
𝑥+𝑦 4 3 4
3. From the following matrix equation, find the value of 𝑥 =
−5 3𝑦 −5 6
3𝑥 − 2𝑦 5 3 5
4. If = , then find the value of y
𝑥 −2 −3 −2
5. Write a square matrix of order 2, which is both symmetric and skew symmetric.
6. If matrix 𝐴 = 1,2,3 , then write AA, where A is the transpose of matrix A
0 𝑎 −3
7. If the matrix 𝐴 = 2 0 −1 is skew symmetric, find the value of ‘a’ and ‘b’
𝑏 1 0
8. If A is a square matrix such that 𝐴2 = 𝐴, then write the value of 𝐼 + 𝐴 3 − 7𝐴
9. If a matrix has 5 elements, write all possible orders it can have.
𝑖−𝑗
10. Write the elements 𝑎23 𝑜𝑓 𝑎 3 × 3 matrix 𝐴 = 𝑎𝑖𝑗 whose elements 𝑎𝑖𝑗 are given by 𝑎𝑖𝑗 =
2
11. Write the number of all possible matrices of order 2x2 with each entry 1,2 or3.
Very short questions
−2 2 0 2 0 −2
12. Find a matrix A such that 2A – 3B + 5C = 0, where B = and C =
3 1 4 7 1 6
0 2 0 3𝑎
13. If A = and kA = , then find the value of k, a and b
3 −4 2𝑏 24
4 −3
14. Express A = as a sum of a symmetric and a skew-symmetric matrix.
2 −1
1 0
15. Solve the following matrix equation for 𝑥: 𝑥 1 =0
−2 0
3 4 1 𝑦 7 0
16. If 2 + = , find (x - y)
5 𝑥 0 1 10 5
2 4 −2 5
17. If A = and B = , then find (3A – B).
3 2 3 4
2 3 1 −3 −4 6
18. If = , then write the value of x
5 7 −2 4 −9 𝑥
3 −3
19. If matrix A= and 𝐴2 =A, then write the value of.
−3 3
2 −2
20. If matrix A = and 𝐴2 = pA, then write the value of p.
−2 2
1 2
21. Given matrix 𝐴 = , find 𝑓 𝐴 , 𝑖𝑓 𝑓 𝑥 = 2𝑥 2 − 3𝑥 + 5
3 4
1 0 2
22. If 𝐴 = 0 2 1 , then show that 𝐴3 − 6𝐴2 + 7𝐴 + 2𝐼 = 0
2 0 3
2 3 1
23. Express the matrix 1 −1 2 as the sum of a symmetric and a skew symmetric matrix.
4 1 2
2 0 1
24. If 𝐴 = 2 1 3 . Then find the value of 𝐴2 − 3𝐴 + 2𝐼
1 −1 0
25. Show that the elements along the main diagonal of a skew symmetric matrix are all zero.
0 6 7 0 1 1 2
26. If 𝐴 = −6 0 8 , 𝐵 = 1 0 2 , 𝐶 = −2 , then calculate AC,BC, and (A+B)C. Also verify that
7 −8 0 1 2 0 3
(A+B)C=AC+BC
27. A manufacturer produces three products 𝑥, 𝑦, 𝑧 which he sells in two markets. Annual sales are indicated in
the table
Market Products
X Y Z
I 10,000 2,000 18,000
II 6,000 20,000 8,000
If unit sale price of x,y and z are Rs. 2.50 ,Rs.150 and Rs.1.00 respectively, then find the total revenue each
market, using matrices.
Multiple choice questions.
28. Choose and write the correct option in each of the following questions.
𝛼 3 4
(i) 𝐼𝑓 1 2 1 =0 then the value of 𝛼 is
1 4 1
(a) 1 (b) 2 (c) 3 (d) 4
(ii) If A,B are non-singular square matrices of the same order, the (𝐴𝐵−1 )=
(a) 𝐴−1 𝐵 (b) 𝐴−1 𝐵−1 (c) 𝐵𝐴−1 (d) AB
2 3
(iii) If A = be such that 𝐴−1 = kA , then k equals
5 −2
1 −1
(a) 19 (b) (c)-19 (d)
19 19
1 −𝑡𝑎𝑛𝜃 1 𝑡𝑎𝑛𝜃 −1 𝑎 −𝑏
(iv) If = , then
𝑡𝑎𝑛𝜃 1 −𝑡𝑎𝑛𝜃 1 𝑏 𝑎
(a) a=1,b=1 (b) a= cos 2𝜃, b = sin 2𝜃 (c) a= sin 2𝜃 b= cos 2𝜃 (d) None of these
(v) If A is a square matrix of order 3, such that A(adj)A)=10 I, then |adj A| is equal to
(a) 1 (b) 10 (c) 100 (d) 101
(vi) If A satisfies the question 𝑥 − 5𝑥 + 4𝑥 +  = 0 then 𝐴−1 exists if
3 2

(a)  ≠ 1 (b)  ≠ 2 (c)  ≠ −1 (d) all of them

(vii) If A is a square matrix of order 3 and |A| =5 , then |adj A| =
(a) 20 (b) 25 (c) 0 (d) 100
2 −3 5
29. If 𝐴𝑖𝑗 is the cofactor of the element 𝑎𝑖𝑗 of the determinant 6 0 4 , then write the value of 𝑎32. 𝐴32.
1 5 7
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
30. If𝐴 = , then for any natural number n, find the value of det(A”)
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
1 −2
31. Find the cofactors of all the elements of
4 3
2 −1
32. Write the adjoint of the matrix
4 3
33. If A is a square matrix of order 3 such that |adjA| =64 find |A|
𝑎 𝑏 1 0
34. If 𝐴 = and 𝐵 = then find adj (AB)
𝑐 𝑑 0 1
0 −1 3 5
35. Find |AB| , if 𝐴 = and 𝐵 =
0 2 0 0
36. Let A be a square matrix of order 3 x 3 . Write the value of |2A| where |A| =4
37. A is a square matrix of order n with |A| =4. Then find the value of |A (adjA)|
3 −4
38. For A = write 𝐴−1
1 −1
39. Find the equation of lie joining (3,1) and (9,3) using determinant.
1 𝑥 𝑦𝑧
40. Using co-factors of elements of third column, evaluate ∆ = 1 𝑦 𝑧𝑥
1 𝑧 𝑥𝑦
2 3 1
41. If A = , show that 𝐴−1 = 19 𝐴
5 −2
42. Solve the following system of equations by matrix method. 3𝑥 + 𝑦 = 19 3𝑥 − 𝑦 = 23
43. Show that the points 𝐴 𝑎, 𝑏 + 𝑐 , 𝐵 𝑏, 𝑐 + 𝑎 , 𝐶(𝑐, 𝑎 + 𝑏) are collinear.
44. A typist charges Rs.145 for typing 10 english and 3 Hindi pages. While charges for typing 3 English and 10
Hindi pages are Rs.180. Using matrices, find the charges of typing one English and one hindi pages
separately.
45. If A,B are square matrices of the same order, then prove that adj(AB) =(adjB)(adjA)
2 3 4 −6
46. Let A = B= . Then compute AB. Hence solve the following system of equations 2𝑥 + 𝑦 = 4,
1 2 −2 4
3𝑥 + 2𝑦 = 1.
3 7 6 8
47. Let A = B= , verify that 𝐴𝐵 −1 = 𝐵−1 𝐴−1
1 5 7 9
0 2 −3
48. Without using properties of determinant prove that −2 0 4 =0
3 −4 0
1+𝑥 1 1
49. Without using properties of determinant prove that 1 1+𝑦 1 = 𝑥𝑦𝑧 + 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥
1 1 1+𝑧
2 −3 5
50. If 𝐴 = 3 2 −4 find 𝐴−1 , use it to solve the system of equations
1 1 −2
2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2𝑦 − 4𝑧 = −5, 𝑥 + 𝑦 − 2𝑧 = −3
1 1 1
51. If 𝐴 = 1 0 2 , Find 𝐴−1 . Hence, solve the system of equations
3 1 1
𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑧 = 7, 3𝑥 + 𝑦 + 𝑧 = 12
52. Using matrix, solve the following system of equations:
2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2𝑦 − 4𝑧 = −5, 𝑥 + 𝑦 − 2𝑧 = −3
−1 −2 −2
53. Find the adjoint of the matrix 𝐴 = 2 1 −2 and hence ,show that A. (AdjA)=|A|𝐼3
2 −2 1
3 −1 1
54. Find the inverse of matrix 𝐴 = −15 6 −5 and hence show that 𝐴−1 , 𝐴 = 𝐼.
5 −2 2
55. Using matrices, solve the following system of linear equations
3𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 1, 4𝑥 − 3𝑦 + 2𝑧 = 4
56. Using matrices, solve the following system of equations
𝑥 + 2𝑦 + 𝑧 = 7, 𝑥 + 3𝑧 = 11, 2𝑥 − 3𝑦 = 1
57. Using matrices, solve the following system of equations
𝑥 − 𝑦 + 𝑧 = 4, 2𝑥 + 𝑦 − 3𝑧 = 0, 𝑥+𝑦+𝑧 =2
58. Using matrices, solve the following system of equations
2𝑥 + 3𝑦 + 3𝑧 = 5, 𝑥 − 2𝑦 + 𝑧 = −4, 3𝑥 − 𝑦 − 2𝑧 = 3
59. Using matrices, solve the following system of equations
𝑥 + 𝑦 − 𝑧 = 3, 2𝑥 + 3𝑦 + 𝑧 = 10, 3𝑥 − 𝑦 − 7𝑧 = 1
1 2 0
60. If 𝐴 = −2 −1 −2 ,find𝐴−1 Hence solve the system of equations
0 −1 1
𝑥 − 2𝑦 = 10, 2𝑥 − 𝑦 − 𝑧 = 8, −𝑦 + 𝑧 = 7
1 −1 0 2 2 −4
61. Evaluate the product AB ,where 𝐴 = 2 3 4 𝑎𝑛𝑑 𝐵 = −4 2 −4
0 1 21 2 −1 5
62. Hence solve the system of linear equations
𝑥 − 𝑦 = 3, 2𝑥 + 3𝑦 + 4𝑧 = 17, 𝑦 + 2𝑧 = 7