Unit-I (2marks questions) - the characteristic equation of the matrix 1 2 0 2 and get its...

download Unit-I (2marks questions) - the characteristic equation of the matrix 1 2 0 2 and get its eigenvalues. Sol. Given is a upper triangular matrix. Hence the eigenvalues are 1,2 W.k.t

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  • UNIT-I (2marks questions)

    1. Find the characteristic equation of the matrix 1 20 2

    .

    Sol. The characteristic equatin of A is 0A I =

    2

    2

    1 2 1 00

    0 2 0 1

    1 20

    0 2(1 )(2 ) 0 0

    2 2 0

    3 2 0

    =

    =

    = + =

    + = The required characteristic equation is 2 3 2 0 + = .

    2. Obtain the characteristic equation of 1 25 4

    .

    Sol.

    Let A=1 25 4

    The characteristic equation of A is 2 1 2 0c c + =

    1

    2

    1 4 5

    1 25 4

    4 106

    c sumof the maindiagonal elements

    c A

    == + ==

    =

    = =

    2

    2

    (5) ( 6) 0

    5 6 0

    Hencethecharacteristic equationis

    + = =

    3. Find the sum and product of the eigenvalues of the matrix 1 1 1

    1 1 11 1 1

    .

    Sol.

    ( 1) ( 1) ( 1)3

    1 1 11 1 11 1 1

    1(1 1) 1( 1 1) 1(1 1)1(0) 1( 2) 1(2)

    4

    sumof theeigenvalues sum ofthe diagonal elements

    product of theeigenvalues

    == + + =

    =

    = + += +=

    4. Two eigen values of the matrix

    11 4 77 2 5

    10 4 6

    are 0 and 1,

    find the third eigen value. Sol.

    1 2 3

    1 2 3

    3

    3

    0, 1, ?

    11 ( 2) ( 6)0 1 3

    2

    Givensumof theeigenvalues sum of the main diagonal elements

    = = ==

    + + = + + + + =

    =

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  • 5. Verify the statement that the sum of the elements in the diagonal of a matrix is the sum of the eigenvalues of the matrix

    2 2 32 1 61 2 0

    .( 2) (1) (0)

    12 2 3

    2 1 61 2 0

    2(0 12) 2(0 6) 3( 4 1)24 12 945

    sol sum of theeigenvalues sumof the maindiagonal elements

    product of theeigenvalues

    == + +=

    =

    = += + +=

    6. The product of the eigenvalues of the matrix

    6 2 22 3 1

    2 1 3A

    =

    is 16, Find the third eigenvalue. Sol.

    1, 2 3

    1 2

    1 2 3

    , .

    16

    6 2 22 3 1

    2 1 3

    let theeigenvalues of the matrix Abe

    Givenwe knowthat A

    =

    =

    =

    6(9 1) 2( 6 2) 2(2 6)6(8) 2( 4) 2( 4)

    = + + + = + +

    3

    3

    3216 32

    2

    ===

    7. Two eigenvalues of the matrix

    1 2 3

    1 2 3

    3

    3

    1 2

    8 6 26 7 4 3 0. ?

    2 4 3

    . 3, 0, ?. .

    8 7 33 0 18

    15

    are and what is the product of theeigenvalues of A

    sol givenw k tThe sum of theeigenvalues sumof the main diagonal elements

    productofeigenvalues

    = = ==

    + + = + ++ + =

    == 3 (3)(0)(15) 0 = =

    8. Find the sum and product of the eigen values of the matrix 2 0 10 2 01 0 2

    .2 2 26

    2 0 10 2 01 0 2

    2(4 0) 0(0) 1(0 2)8 26

    sol sumof theeigenvalues sum of the main diagonal elements

    product of theeigenvalues A

    == + +==

    =

    = + = =

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  • 9.Find the characteristic equation of the matrix 1 20 2

    and get its

    eigenvalues.Sol. Given is a upper triangular matrix. Hence the eigenvalues are 1,2 W.k.t the chacteristic equation of the given matrix is

    2

    2

    2

    ( ) ( ) 0

    (1 2) (1)(2) 0

    3 2 0

    sumof theeigenvalues product of theeigenvalues

    + = + + = + =

    10.Prove that if is an eigenvalues of a matrix A, then 1

    is the

    eigenvalue of 1A

    1

    1 1

    1

    1

    1

    1

    ;

    ,

    1

    1. ,

    proofIf X betheeigenvector corresponding to

    then AX X

    premultiplying bothsides by A weget

    A AX A X

    IX A X

    X A X

    X A X

    i e A X X

    =

    ===

    =

    =

    11.Find the eigenvalues of A given

    3 3 3

    1 2 30 2 70 0 3

    .1 2 30 2 70 0 3

    1,2,31,2,3

    1 ,2 ,

    A

    sol

    A

    clearly given Ais aupper triangular matrixHencetheeigenvalues aretheeigenvalues of the given matrix Aare

    By the property theeigenvalues of the matrix A are

    =

    =

    33 .

    12.If and are cthe eigen values of 3 11 5

    form the

    matrix whose eigenvalues are 3 and 3

    Sol.

    2 2 2

    1 7 50 2 9 00 0 5

    (1 )[(2 )(5 ) 0] 7[0 0] 5[0 0] 0(1 ) (2 )(5 ) 0

    1, 2, 5

    1 2 530

    sumof theeigenvalues

    =

    + =

    == = =

    = + +=

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  • 13.Sum of square of the eigenvalues of

    1 7 50 2 90 0 5

    is..

    Sol. The characteristic equatin of A is 0A I =

    2 2 2

    1 7 50 2 9 00 0 5

    (1 )[(2 )(5 ) 0] 7[0 0] 5[0 0] 0(1 ) (2 )(5 ) 0

    1, 2, 5

    1 2 530

    sumof theeigenvalues

    =

    + =

    == = =

    = + +=

    14 .two eigenvalues of A=

    4 6 61 3 21 5 2

    are equal and they are

    double the third.Find the eigenvalues of A. Sol.

    2 2 2 2

    2 ,2

    2 2 (4) (3) ( 2)5 5

    12,2,1

    2 ,2 ,1

    LetthethirdeigenvaluebeTheremainingtwoeigenvaluesare

    sumftheeigenvalues sumofthemaindiagonalelements

    theeigenvaluesofAare

    HencetheeigenvaluesofA are

    =+ + = + +

    ==

    15.show that the matrix 1 22 1

    satisfies its own characteristic

    equation.

    Sol.

    1 22 1

    LetA

    =

    The cha.equation of the given matrix is

    21 2

    1

    2

    2

    2

    2

    2

    0

    0

    1 1 21 2

    1 4 52 1

    2 5 0

    2 5 0

    .1 2 1 22 1 2 1

    3 44 3

    3 4 1 2 1 02 5 2 5

    4 3 2 1 0 1

    0 00 0

    A I

    S SS sumof main iagonal elements

    S A

    Thecharacteristic is

    Toprove A A I

    A A A

    A A I

    =

    + === + =

    = = = + =

    + = + ==

    =

    =

    + = +

    =

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  • 16.If A=1 04 5

    express 3A in terms of A and I using Cayley

    Hamilton theorem.Sol.The cha.equation of the given matrix is 0A I =

    2

    1 0 1 00

    4 5 0 1

    1 00

    4 5(1 )(5 ) 0 0

    (1 )(5 ) 0

    6 5 0

    =

    =

    = =

    + =

    By Cayley Hamilton theorem,

    2 2

    3 2

    3 2

    6 5 0, 6 5

    6 5 0

    6 56(6 5 ) 536 30 531 30

    A A I A A Imultiply Aon both sides

    A A A

    A A AA I A

    A I AA I

    + = =

    + == = = =

    17.Write the matrix of the quadratic form 2 22 8 4 10 2x z xy xz yz+ + .

    Sol.

    Q=

    2

    2

    2

    1 12 2

    1 12 21 12 2

    coeff of x coeff of xy coeff of xz

    coeff of xy coeff of y coeff of yz

    coeff of xz coeff of yz coeff of z

    Q=

    2 2 52 0 15 1 8

    18.Determine the nature of the following quadratic form

    ( ) 2 21 2 3 1 2, , 2. .

    f x x x x x

    sol The matrix of Q F is

    = +

    Q=2

    2

    2

    1 12 2

    1 12 21 12 2

    coeff of x coeff of xy coeff of xz

    coeff of xy coeff of y coeff of yz

    coeff of xz coeff of yz coeff of z

    =

    1 0 00 2 00 0 0

    There for the eigenvalues are 0,1,2. so find the eigenvalues one eigenvalue is Zero another two eigenvalues are positive .so given Q.F is positive semi definite.

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  • 19. State Cayley Hamilton theorem. Every square matrix satisfies its own characteristic equation.

    20. Prove that the Q.F 2 2 22 3 2 2 2x y z xy yz zx+ + + + .

    Sol.The matrix of the Q.F form,

    Q=

    2

    2

    2

    1 12 2

    1 12 21 12 2

    coeff of x coeff of xy coeff of xz

    coeff of xy coeff of y coeff of yz

    coeff of xz coeff of yz coeff of z

    =

    1 1 11 2 11 1 3

    1 1

    1 12

    2 2

    1 1 1

    3 2 2 2

    3 3 3

    1 1( )

    1 1(2 1) 1( )

    1 2

    1(6 1) 1(3 1) 1(1 2) 2( )

    D a ve

    a bD ve

    a b

    a b cD a b c ve

    a b c

    = = = +

    = = = = +

    = = + + =

    The Q.F is indefinite.

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  • UNIT II - SEQUENCES AND SERIESPart A

    1. Given an example for (i) convergent series (ii) divergent series (iii) oscillatory series

    Solution:

    (i) The series

    + is convergent

    (ii) 1+2+3+.+n+ is divergent

    (iii) 1-1+1-1+ is oscillatory

    2. State Leibnitzs test for the convergence of an alternating series

    Solution:

    The series a1-a2+a3-a4+. In which the terms are alternately +ve and ve and all ais are positive, is convergent if

    (i) and

    (ii)

    3. State the comparison test for convergence of series

    Solution:

    Let an and bn be any two series and let a

    finite quantity 0, then the two series converges or diverges together

    4. State any two properties of an infinite series

    Solution:

    (i) The converges or diverges of an infinite series is not affected when each of its terms is multiplied by a finite quantity

    (ii) If a series in which all the terms are positive is convergent, the series will remain convergent even when some or all of its terms are made negative

    5. Define alternating series

    Solution:

    A series whose terms are alternatively positive and negative is called alternating s