LOYOLA ICAM COLLEGE OF ENGINEERING AND TECHNOLOGY
Loyola College Campus, Nungambakkam, Ch-34
MATHEMATICS-I (MA 6151)
PART-A
UNIT-I
EIGEN VALUES & EIGEN VEECTORS
6 −2 2
1. The product of two Eigen values of the matrix 𝐴 = (−2 3 −1) is 16. Find the third eigen value of
2 −1 3
A. (AU JAN ’12)
0
2. Find the Eigen values of A = (0
3
0
2
−1 4 ) . Also find the Eigen values of -3A.
1 −5
1
3. For what values of ‘c’ the Eigen values of the matrix (
c
complex conjugates?
−1 0 0
4. Find the Eigen values of A2 given 𝐴 = ( 2 −3 0).
1
4 2
(AU N/D ’11)
2
) are real and unequal, real and equal,
4
(AU Jan ’10)
5. If 1 and 2 are the Eigen values of a 2 X 2 matrix A, what are the Eigen values of 𝐴2 and 𝐴−1 .
(AU M/J ’10)
1 −2
6. If -1 is an Eigen value of the matrix 𝐴 = (
), find the eigen values of A 4 using properties.
−3 2
(AU N/D ’10)
7. For a given matrix A of order 3, |A| = 32 and two of its eigen values are 8 and 2,
find the third.(AU JAN ’09)
8. If the sum of two eigen values and trace of a 3 X 3 matrix A are equal, Find the value of |A|.
(AU M/J ’09)
9. If the sum of two eigen values and trace of a 3 X 3 matrix A are equal, find |𝐴|. (AU A/M ’08)
10. Find the sum and product of the eigen values of the square matrix using the properties
8 1 6
A = (3 5 7) .
4 9 2
(AU A/M ’04)
1
11. Find the symmetric matrix A, whose eigen values are 1 and 3 with corresponding eigen vectors (−1
)
and ( 11 ).
(AU JAN ’13)
12. If the eigen values of a matrix A of order 3 X 3 are 2, 3 and 1, then find the eigen values of adj(A).
(AU JAN ’14)
13. If λ is the eigen value of A , then prove that λ2 is the eigen value of A2 . (AU JAN ’14)
2 1 0
14. Find the eigen values of the inverse of the matrix 𝐴 = (0 3 4). (AU JAN ’14)
0 0 4
15. If 2, -1, -3 are the eigen values of the matrix A, then find the eigen values of 𝐴2 − 2𝐼. (AU JAN ’14)
DIAGONALISATION (ORTHOGONAL REDUCTION)
1 0
16. Can 𝐴 = (
) be diagonalized? Why?(AU JAN ’12)
0 1
17. Find the nature of the quadratic form 𝑥1 2 + 2𝑥2 2 + 𝑥3 2 − 2𝑥1 𝑥2 + 2𝑥2 𝑥3 .
(AU N/D ’11)
1 0
18. Can 𝐴 = (
) be diagonalized ?why ?(AU Jan ’10)
0 1
cos θ
19. Check whether the matrix B is orthogonal? Justify B = (−sin θ
0
sinθ 0
cos θ 0) . (AU JAN ’09)
0
1
20. If A is an orthogonal matrix, prove that A Tand A −1 are orthogonal matrices.(AU N/D ’07)
21. If A is an orthogonal matrix, show that A−1 is also orthogonal.
(AU A/M ’04)
2 0 −2
22. Write down the quadratic form corresponding to the matrix [ 0 2 1 ]. (AU JAN ’13)
−2 1 −2
CAYLEY HAMILTON THEOREM
23. Give two uses of Use Cayley-Hamilton theorem. (AU A/M ’08)
5
24. Use Cayley-Hamilton theorem to find 𝐴4 − 8𝐴3 − 12𝐴2 when 𝐴 = (
1
3
).
3
(AU N/D ’10)
1 2
25. Use Cayley-Hamilton theorem to find (𝐴4 − 4𝐴3 − 5𝐴2 + 𝐴 + 2𝐼) when = (
) . (AU M/J ’09)
4 3
26. State Cayley-Hamilton theorem.
(AU N/D ’07)
PART-B
EIGEN VALUES & EIGEN VECTORS
2 2 1
1. Find the eigen values and eigen vectors of the matrix A = (1 3 1) .
1 2 2
(AU N/D ’07& M/J ’10& N/D’10& Jan ’12 & Jan ’14 ).
2 1
2. Find the eigen values and eigen vectors of A = (1 2
0 0
1
1).
1
(AU Jan ’09).
−2 2 −3
3. Find theeigen values and eigen vectors for the matrix 𝐴 = ( 2
1 −6). (AU M/J ’09).
−1 −2 0
1 −1 4
4. Find all the eigen values and eigen vectors of the matrix A = (3 2 −1). (AU Jan ‘ 11).
2 1 −1
−2 2 −3
5. Find the eigen values and eigen vectors of A = ( 2
1 −6) . (AU Jan ‘ 10 & AU M/J 14).
−1 −2 0
4 1 1
6. Find the eigen values and eigen vectors of the matrix A = (1 4 1) . (AU M/J ’07)
1 1 4
11 −4 −7
7. Find the eigen values and eigen vectors of the matrix = ( 7 −2 −5) . (AU N/D ’11)
10 −4 −6
1
2
1
8. Find the eigen values and eigen vectors of the matrix = ( 6 −1 0 ) . (AU Jan ‘ 13).
−1 −2 −1
9. Prove that the eigen values of a real symmetric matrix are real. (AU Jan ’14)
CAYLEY HAMILTON THEOREM
1 0
3
10. Using Cayley- Hamilton theorem , find 𝐴4 if A = (2 1 −1) . (AU N/D ’07).
1 −1 1
1 0
3
11. Verify Cayley – Hamilton theorem for the matrix (2 1 −1) and find 𝐴−1. (AU M/J 14).
1 −1 1
2 −1 1
12. Find the Characteristic equation of the matrix A given A = (−1 2 −1). Hence find
1 −1 2
𝐴4 𝑎𝑛𝑑 𝐴−1 . (AU Jan ’09).
2 −1 2
13. Using Cayley- Hamilton theorem , find 𝐴−1 𝑤ℎ𝑒𝑛 𝐴 = (−1 2 −1).
1 −1 2
(AU M/J ’10).
14. Use Cayley- Hamilton theorem to find the value of the matrix given by
2 1 1
𝐴8 − 5𝐴7 + 7𝐴6 − 3𝐴5 + 𝐴4 − 5𝐴3 + 8𝐴2 − 2𝐴 + 𝐼, if the matrix A = (0 1 0). (AU M/J ’09).
1 1 2
15. Find 𝐴−1 for the matrix
1 2
A = (2 4
3 5
3
5) using Cayley- Hamilton theorem.
6
(AU A/M ’08).
−1 0 3
16. Using Cayley- Hamilton theorem find the inverse of the matrix A = ( 8 1 −7).
−3 0 8
(AU A/M ’04 & N/D ‘10).
1 −1 1
17. Show that the matrix A = (0 1 0) satisfies the characteristic equation and hence find its
2 0 3
inverse. (AU Jan ‘ 11 &AU Jan ‘ 11)
2 −1 1
18. Verify Cayley Hamilton theorem and hence find 𝐴−1 for A = (−1 2 −1). (AU Jan ‘ 10).
1 −1 2
2 −1
19. Using Cayley- Hamilton theorem find 𝐴−1 and 𝐴3 + 𝐴6 , if 𝐴 = (
) . (AU M/J ’07)
5 −2
−1 0 3
20. Using Cayley- Hamilton theorem find 𝐴−1 of the matrix 𝐴 = ( 8 1 7) . (AU N/D ’10)
−3 0 8
1
21. Using Cayley- Hamilton theorem find 𝐴−1 of the matrix 𝐴 = (4
1
3 7
2 3).
2 1
(AU N/D ’11)
1 4
22. Find 𝐴𝑛 using Cayley Hamilton theorem, taking 𝐴 = (
). Hence find 𝐴3 . (AU Jan ’12)
2 3
−1
23. Using Cayley- Hamilton theorem find 𝐴
1
2 −2
and 𝐴 if 𝐴 = (−1 3
0 ). (AU Jan’14)
0 −2 1
4
DIAGONALISATION TO CANONICAL FORM BY ORTHOGONAL REDUCTION
24. Reduce the quadratic form 𝑥1 2 + 2𝑥2 2 + 𝑥3 2 − 2𝑥1 𝑥2+2𝑥2 𝑥3 to canonical form through an orthogonal
transformation and hence show that is positive semi definite. Also given a non-zero set of values
(𝑥1 , 𝑥2 , 𝑥3 ) which makes this quadratic form zero.(AU M/J ’09).
25. Reduce the quadratic form below to its Canonical form by an orthogonal reduction
3 𝑥1 2 + 2𝑥2 2 + 3𝑥3 2 − 2𝑥1 𝑥2 − 2𝑥2 𝑥3 .(AU N/D ’07).
26. Reduce the quadratic form 2𝑥 2 + 5𝑦 2 + 3𝑧 2 + 4𝑥𝑦 to canonical form by orthogonal reduction and
state its nature. (AU M/J ’10).
1
27. Diagonalize the matrix A = (0
0
0
0
3 −1) using an orthogonal transformation. (AU A/M ’08).
−1 3
10 −2 −5
28. Reduce the matrix A = (−2 2
3 ) to diagonal form. (AU A/M ’04).
−5 3
5
29. Find a change of variables that reduces the quadratic form 3𝑥1 2 + 5𝑥2 2 + 3𝑥3 2 − 2𝑥1 𝑥2+2𝑥1 𝑥3 −
2𝑥2 𝑥3 to a sum of squares and express the quadratic form in terms of new variables. (AU Jan ‘ 11).
30. Reduce the quadratic form 10𝑥1 2 + 2𝑥2 2 + 5𝑥3 2 + 6𝑥2 𝑥3 − 10𝑥3 𝑥1 − 4𝑥1 𝑥2
to a canonical form
through an orthogonal transformation and hence find rank, index, signature, nature and also give the
non-zero set of values for 𝑥1 , 𝑥2 , 𝑥3 .(if they exist) , that will make the quadratic form zero. (AU Jan ‘ 10).
31. If the quadratic form 3𝑥 2 + 6𝑦 2 + 3𝑧 2 − 4𝑥𝑦 + 4𝑦𝑧 − 2𝑧𝑥 is reduced to the canonical form
𝑐1 𝑦1 2 + 𝑐2 𝑦2 2 + 𝑐3 𝑦3 2 by an orthogonal transformation, find the constansts𝑐1 , 𝑐2 , 𝑐3 . Also state the nature
of the quadratic form. (AU M/J ’07)
32. Reduce the quadratic form 2𝑥1 2 + 𝑥2 2 + 𝑥3 2 − 4𝑥2 𝑥3 − 2𝑥3 𝑥1 + 2𝑥1 𝑥2 to a canonical form through
an orthogonal transformation and hence find rank, index, signature and nature. (AU N/D ’10)
33. Reduce the quadratic form 8 𝑥1 2 + 7𝑥 2 + 3𝑥3 2 − 8𝑥2 𝑥3 + 4𝑥3 𝑥1 + 12𝑥1 𝑥2 into
means of an orthogonal transformation. (AU N/D ’11)
canonical form by
34. Reduce the quadratic form 2𝑥 2 + 5𝑦 2 + 3𝑧 2 + 4𝑥𝑦 to canonical form by orthogonal reduction and
state its nature.(AU Jan ’12)
35. Reduce the quadratic form 2𝑥1 𝑥2 + 2𝑥3 𝑥2 + 2𝑥1 𝑥3
𝑖𝑛 to canonical form.
(AU Jan ’13)
8 −6 2
36. If the eigen values of [−6 7 −4]
2 −4 3
matrix .
are 0, 3, 15, find the eigen vectors of A and diagonalize the
(AU Jan ’13)
37. Reduce the quadratic form 6𝑥 2 + 3𝑦 2 + 3𝑧 2 − 4𝑥𝑦 − 2𝑦𝑧 + 4𝑧𝑥 into an canonical form by orthogonal
reduction. Hence find its rank and nature. (AU Jan ’14)
38. Reduce the quadratic form 8 𝑥1 2 + 5𝑥 2 + 𝑥3 2 + 2𝑥2 𝑥3 + 6𝑥3 𝑥1 + 2𝑥1 𝑥2 into canonical form by
orthogonal transformation and find its nature. (AU Jan ’14)