MA6151 Part B 16 Marks
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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
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