KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
Unit I
Part – A
1.
Define Eigen vector.
[CO1, K1]
2.
Define Eigen value.
[CO1, K1]
3.
What do you mean by Characteristic equation of a matrix?
[CO1, K1]
4.
State any two properties of Eigen values.
[CO1, K1]
5.
List some of the properties of Eigen vectors.
[CO1, K1]
6.
2
Find the sum and product of the eigen values of the matrix A = [1
2
1 2
3 1 ].
1 −6
[CO1, K2]
7.
3
Find the sum and product of the eigen values of the matrix A = [0
1
1 2
5 6].
6 3
[CO1, K2]
8.
For a given matrix 𝐴 of order 3, |𝐴| = 32 and two of its Eigen values are 8 and 2, find the sum of
[CO1, K2]
Eigen values.
9.
6 −2 2
The product of two Eigen values of the matrix A = [−2 3 −1] is 16. Find the third Eigen
2 −1 3
[CO1, K2]
value.
10.
7
One of the Eigen values of A = [4
4
4
4
−8 −1] is -9. Find the other two Eigen value.
−1 −8
11.
2 5
If 𝐴 = [0 3
0 0
12.
4
6
6
Two Eigen values of 𝐴 = [ 1
3
2 ] are equal and they are double the third. Find the
−1 −5 −2
−1
2 ], find the eigen values of 𝐴−1 .
4
[CO1, K2]
[CO1, K2]
[CO1, K2]
Eigen values of 𝐴−1 .
13.
1
Find the sum of the squares of the Eigen values of [0
0
2 3
4 5].
0 6
[CO1, K2]
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
14.
1
Two of the Eigen values of A = [2
2
2 2
1 2] are 1 and -1. Find the Eigen values of A−1 .
2 1
15.
2 2 0
Two of the Eigen values of [ 2 1 1 ] are -4 and 3. Find the third Eigen value.
−7 2 −3
[CO1, K2]
16.
2
If 3 and -4 are two eigen values of the matrix 𝐴 = [2
0
[CO1, K2]
2 −7
1 2 ], find the sum and product of the
1 −3
[CO1, K2]
eigen values of the matrix −2𝐴2 .
17.
8 −6 2
If 3 and 15 are the two eigen values of [−6 7 −4]. Find the value of |A|.
2 −4 3
[CO1, K2]
18.
2
7 −2 0
If 𝑋 = [ 1 ] is the Eigen vector of the matrix 𝐴 = [−2 6 −2], find the corresponding Eigen
−2
0 −2 5
[CO1, K2]
value.
19.
2
6 −2 2
If 𝑋 = [−1] is the Eigen vector of the matrix 𝐴 = [−2 3 −1], find the corresponding eigen
1
2 −1 3
[CO1, K2]
value.
20. If the sum of two eigen values and trace of a 3 × 3 matrix A are equal, find |A|.
[CO1, K2]
21. If the eigen values of a matrix 𝐴 are 2, 3, 4. Find the eigen values of 𝐴−1 and 𝐴3 .
[CO1, K2]
22.
3
If A = [0
0
1 4
2 6], find the eigen values of A3 , A−1.
0 5
[CO1, K2]
23. State Cayley-Hamilton theorem.
[CO1, K1]
24. Write the applications of Cayley-Hamilton theorem.
[CO1, K1]
25. Verify Cayley-Hamilton theorem for A = [5 3].
1 3
[CO1, K2]
26. Using Cayley-Hamilton theorem, find the inverse of [1
2
27. If 𝐴 = [1 2] ,find 𝐴−1 using Cayley-Hamilton theorem.
3 4
4
].
3
[CO1, K2]
[CO1, K2]
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
28.
1
If 𝐴3 − 3𝐴2 − 𝐴 + 𝐾𝐼 = 0 satisfies the matrix 𝐴 = [2
1
0
3
1 −1], find 𝐾.
−1 1
[CO1, K2]
29. Define orthogonal matrix.
[CO1, K1]
30. Show that [ 𝑐𝑜𝑠𝜃
−𝑠𝑖𝑛𝜃
[CO1, K2]
𝑠𝑖𝑛𝜃
] are orthogonal.
𝑐𝑜𝑠𝜃
31. State any two properties of orthogonal matrices.
[CO1, K1]
32. If A is an orthogonal matrix, then prove that AT is also an orthogonal matrix.
[CO1, K1]
33. If A is an orthogonal matrix, then prove that A−1 is also an orthogonal matrix
[CO1, K1]
34. If A and B are orthogonal matrices, then prove that AB is also an orthogonal matrix.
[CO1, K1]
35. For an orthogonal matrix, prove that |A| = ±1
[CO1, K1]
36. Prove that a square matrix is orthogonal if and only if A−1 = AT .
[CO1, K1]
37. Define Similarity Transformation.
[CO1, K1]
38. What do you mean diagonalization of a matrix?
[CO1, K1]
39. Define orthogonal reduction.
[CO1, K1]
40. How will you diagonalise a square matrix by orthogonal reduction?
[CO1, K1]
41. What is the condition for a matrix to be diagonalised using orthogonal reduction?
[CO1, K1]
42. Define Quadratic form.
[CO1, K1]
43. Define matrix of Quadratic form.
[CO1, K1]
44. Write down the matrix corresponding to the quadratic form 2𝑥 2 + 2𝑦 2 + 3𝑧 2 + 2𝑥𝑦 − 4𝑥𝑧 −
[CO1, K2]
4𝑦𝑧.
45. Write down the matrix of the quadratic form 3𝑥12 + 5𝑥22 + 5𝑥32 − 2𝑥1 𝑥2 + 6𝑥1 𝑥3 + 2𝑥2 𝑥3 .
[CO1, K2]
46. Write the matrix of the quadratic form 3𝑥 2 − 2𝑦 2 − 𝑧 2 + 8𝑧𝑥 + 12𝑦𝑧 − 4𝑥𝑦.
[CO1, K2]
47. Write down the matrix of the quadratic form 2𝑥1 𝑥2 + 4𝑥1 𝑥3 + 6𝑥2 𝑥3 .
[CO1, K2]
48.
2
Find the quadratic form associated with the matrix 𝐴 = [2
0
2 0
5 0].
0 3
[CO1, K2]
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
49.
2
Find the quadratic form associated with the matrix 𝐴 = [5
3
5 3
0 4 ].
4 −1
[CO1, K2]
50. Define canonical form of quadratic form.
[CO1, K1]
51. How will you reduce the given quadratic form to canonical form?
[CO1, K1]
52. Define rank of quadratic form.
[CO1, K1]
53. Define index of quadratic form.
[CO1, K1]
54. Define signature of quadratic form.
[CO1, K1]
55. When do you say that a quadratic form is positive definite?
[CO1, K1]
56. When do you say that a quadratic form is negative definite?
[CO1, K1]
57. When do you say that a quadratic form is positive semi-definite?
[CO1, K1]
58. When do you say that a quadratic form is negative semi-definite?
[CO1, K1]
59. Define indefinite nature of quadratic form?
[CO1, K1]
60. Show that the quadratic form 3𝑥12 + 3𝑥22 + 3𝑥32 + 2𝑥1 𝑥2 + 2𝑥1 𝑥3 − 2𝑥2 𝑥3 is positive definite.
[CO1, K2]
61. Show that the quadratic form 8𝑥12 + 7𝑥22 + 3𝑥32 − 12𝑥1 𝑥2 + 4𝑥1 𝑥3 − 8𝑥2 𝑥3 is positive semi-
[CO1, K2]
definite.
62. Classify the nature of the quadratic form 𝑥 2 + 4𝑥𝑦 + 6𝑥𝑧 − 𝑦 2 + 2𝑦𝑧 + 4𝑧 2 .
[CO1, K2]
63. Discuss the nature of the quadratic form 3𝑥 2 − 2𝑥𝑦 − 6𝑥𝑧 + 3𝑦 2 − 6𝑦𝑧 − 5𝑧 2 .
[CO1, K2]
64. Find the nature of the quadratic form 6𝑥 2 + 3𝑦 2 + 3𝑧 2 − 4𝑥𝑦 + 4𝑥𝑧 − 2𝑦𝑧.
[CO1, K2]
65. Find the nature of the quadratic form 5𝑥 2 + 5𝑦 2 + 14𝑧 2 + 2𝑥𝑦 − 8𝑥𝑧 − 16𝑦𝑧.
[CO1, K2]
66. Find the nature of the quadratic form 𝑥12 − 𝑥22 + 𝑥32 + 4𝑥1 𝑥3 .
[CO1, K2]
Part – B
1.
11
Find the eigen values and eigen vectors of A = [ 7
10
−4 7
−2 −5].
−4 −6
2.
8 −6 2
Find the eigen values and eigen vectors of [−6 7 −4].
2 −4 3
[CO1, K3]
[CO1, K3]
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
3.
1
Find the eigen values and eigen vectors of A = [1
3
1 3
5 1].
1 1
[CO1, K3]
4.
2
Find the eigen values and eigen vectors of A = [1
1
−2 2
1
1 ].
3 −1
[CO1, K3]
5.
2
Find the eigen values and eigen vectors of A = [1
1
2 1
3 1].
2 2
[CO1, K3]
6.
3
Find the eigen values and eigen vectors of A = [1
1
−4 4
−2 4].
−1 3
[CO1, K3]
7.
1
Find the eigen values and eigen vectors of A = [1
2
0 −1
2 1 ].
2 3
[CO1, K3]
8.
3 10 5
Find the eigen values and eigen vectors of A = [−2 −3 −4].
3
5
7
[CO1, K3]
9.
2
Find the eigen values and eigen vectors of A = [0
1
[CO1, K3]
10.
1
Find the characteristic equation of the matrix 𝐴 = [2
1
0 1
2 0].
0 2
0
3
[CO1, K3]
1 −1]. Also compute 𝐴−1 using
−1 1
Cayley-Hamilton theorem.
11.
1
2 −2
Verify Cayley-Hamilton theorem and hence find A−1 , if A = [−1 3
0 ].
0 −2 1
[CO1, K3]
12.
2 −1 1
Verify Cayley-Hamilton theorem for A = [−1 2 −1].Hence find the value of A4 .
1 −1 2
[CO1, K3]
13.
2 0 −1
[CO1, K3]
Verify Cayley-Hamilton theorem for A = [ 0 2 0 ] and hence find the value of A4 and
−1 0 2
A−1 .
14.
2 −1 2
Verify Cayley-Hamilton theorem for A = [4 2 1]. Hence find A4 and A−1 .
1 2 1
[CO1, K3]
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
15.
7
Find the characteristic equation of the matrix 𝐴 = [6
2
−1 3
[CO1, K3]
1 4]. Show that the equation is
4 8
satisfied by 𝐴 and hence find 𝐴−1 and 𝐴4 .
16.
1 1
3
Using Cayley-Hamilton theorem, find the inverse of A, where A = [1 3 −3].
2 −4 −4
[CO1, K3]
17.
1
Using Cayley-Hamilton theorem, find 𝐴−1 if 𝐴 = [2
3
[CO1, K3]
18.
Using Cayley-Hamilton theorem evaluate the value of the matrix 𝐴4 + 𝐴3 − 18𝐴2 − 39𝐴 + [CO1, K3]
2 −2
5 −4].
7 −5
1 2
3
2𝐼, given 𝐴 = [2 −1 4 ].
3 1 −1
19.
If 𝐴 = [
1 2
], use Cayley-Hamilton theorem to express 𝐴6 − 4𝐴5 + 8𝐴4 − 12𝐴3 + 14𝐴2 in [CO1, K3]
−1 3
terms of 𝐴 and I.
20.
2
1 −1
Diagonalize the matrix A = [ 1
1 −2] by means of orthogonal transformation.
−1 −2
1
[CO1, K3]
21.
6 −2 2
Diagonalise the matrix A = [−2 3 −1] by means of orthogonal transformation.
2 −1
3
[CO1, K3]
22.
10 −2 −5
Diagonalise the matrix A = [−2 2
3 ] by means of orthogonal reduction.
−5 3
5
[CO1, K3]
23.
3
Diagonalise the matrix A = [1
1
[CO1, K3]
24.
8 −6 2
Diagonalise the matrix A = [−6 7 −4] by means of orthogonal transformation.
2 −4 3
[CO1, K3]
25.
3 −1 1
Diagonalise the matrix A = [−1 5 −1].
1 −1 3
[CO1, K3]
26.
Reduce the quadratic form 𝑥12 + 2𝑥22 + 𝑥32 − 2𝑥1 𝑥2 + 2𝑥2 𝑥3 to the canonical form through an [CO1, K3]
1
1
3 −1] by means of orthogonal reduction.
−1 3
orthogonal transformation. Also find rank, index, signature and nature of the quadratic form.
KONGU ENGINEERING COLLEGE(AUTONOMOUS)
PERUNDURAI, ERODE – 638060.
I BSc., (Common to All Branches)
22BCC12 Mathematics I- Question Bank
27.
Reduce the quadratic form 2𝑥𝑦 + 2𝑦𝑧 + 2𝑥𝑧 to the canonical form through an orthogonal [CO1, K3]
transformation. Also find rank, index, signature and nature of the quadratic form.
28.
Reduce the quadratic form x12 + 5x22 + x32 + 2x1 x2 + 2x2 x3 + 6x3 x1 to the canonical form [CO1, K3]
through an orthogonal transformation. Also find rank, index, signature and nature of the
quadratic form.
29.
Reduce the quadratic form 3𝑥 2 + 5y 2 + 3z 2 − 2yz + 2zx − 2xy to the canonical form through [CO1, K3]
an orthogonal transformation. Also find rank, index, signature and nature of the quadratic form.
30.
Reduce the quadratic form 6𝑥 2 + 3y 2 + 3z 2 − 4xy − 2yz + 4xz to the canonical form through [CO1, K3]
an orthogonal transformation. Also find rank, index, signature and nature of the quadratic form.
31.
Reduce the quadratic form 8𝑥 2 + 7y 2 + 3z 2 − 12xy − 8yz + 4xz to the canonical form [CO1, K3]
through an orthogonal reduction and hence find rank, index and signature.
32.
Reduce the quadratic form 2x12 + 6x22 + 2x32 + 8x1 x2 to the canonical form through an [CO1, K3]
orthogonal reduction and hence find index, signature and nature.
33.
Reduce the quadratic form 10x12 + 2x22 + 5x32 − 4x1 x2 + 6x2 x3 − 10x3 x1 to the canonical [CO1, K3]
form through an orthogonal transformation and hence find its rank, index, signature and nature.