UNIT-I

PART-B

−2 2 −3

1. Find all the eigenvalues and eigenvectors of the matrix 2 1 −6

−1 −2 0

7 −2 0

2. Find all the eigenvalues and eigenvectors of the matrix −2 6 −2

0 −2 5

3. Find all the eigenvalues and eigenvectors of the matrix

2 2 1

1 3 1

1 2 2

2 −1 2

4. Using Cayley Hamilton theorem find A when A= −1 2 −1

1 −1 2

4

1 2 −2

5. Using Cayley Hamilton theorem find A When A = −1 3 0

0 −2 1

−1

1 0 3

6. Using Cayley Hamilton theorem find A find A = 2 1 −1

1 −1 1

−1

−1 0 3

6. Using Cayley Hamilton theorem find the inverse of the matrix A = 8 1 −7

−3 0 8

1 −1 4

7.Find a A if A = 3 2 −1 , Using Cayley Hamilton theorem.

2 1 −1

−1

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3 1 1

8. Diagonalise the matrix A= 1 3 −1 by means of an orthogonal transformation.

1 −1 3

10 −2 −5

9.Reduse the matrix −2 2 3 to diagonal form.

−5 3 5

3 −1 1

10. Diagonalise the matrix −1 5 −1 by means of an orthogonal

1 −1 3

6 −2 2

11. Diagonalise the matrix −2 3 −1 by an orthogonal

2 −1 3 transformation. 12. Reduce the quadratic form Q = 6 x 2 + 3 y 2 + 3z 2 − 4 xy − 2 yz + 4 zx into canonical form by an orthogonal transformation.

2

2

2

13. Reduce the quadratic form 8 x1 + 7 x2 + 3x3 − 12 x1 x2 − 8 x2 x3 + 4 x3 x1 to the canonical form by an orthogonal transformation and hence show that it is positive semi-definite.

2

2

2

14. Reduce the quadratic form x1 + 5 x2 + x3 + 2 x1 x2 + 2 x2 x3 + 2 x3 x1 to the canonical form by an orthogonal transformation

15. Reduce the quadratic form x 2 + y 2 + z 2 − 2 xy − 2 yz − 2 zx to canonical form by an orthogonal transformation

16. Find all the eigenvalues and eigenvectors of the matrix

8 −6 2

−6 7 −4

2 −4 3

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17.Obtain the orthogonal transformation whish will transform the

Quadratic form Q = 2 x1 x2 + 2 x2 x3 + 2 x3 x1 into sum of squares.

UNIT-II

PART-B

1.

Show that

2.

The series

converges to 0 is convergent and its sum is 1.

3. Prove that the series 1-2+3-4+…. Oscillates