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    Matrices Inter First Year Important

    Uploaded by

    Syed Salman
    1. The document defines key terms related to matrices such as trace, symmetric matrix, skew-symmetric matrix, and provides examples of solving systems of equations using matrices. 2. Examples include finding the trace of a matrix, determining if a matrix is symmetric or skew-symmetric, solving systems of equations using inverse matrices and Cramer's rule, and calculating determinants, adjoints and inverses. 3. Key steps shown are setting up matrices to represent systems of equations, taking the adjoint and inverse, and using the inverse matrix to solve the system of equations.

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    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd

    Matrices Inter First Year Important

    Uploaded by

    Syed Salman
    14K views11 pages
    1. The document defines key terms related to matrices such as trace, symmetric matrix, skew-symmetric matrix, and provides examples of solving systems of equations using matrices. 2. Examples include finding the trace of a matrix, determining if a matrix is symmetric or skew-symmetric, solving systems of equations using inverse matrices and Cramer's rule, and calculating determinants, adjoints and inverses. 3. Key steps shown are setting up matrices to represent systems of equations, taking the adjoint and inverse, and using the inverse matrix to solve the system of equations.

    Original Description:

    inter imp problems

    Original Title

    Matrices inter first year important

    Copyright

    © Attribution Non-Commercial (BY-NC)

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    1. The document defines key terms related to matrices such as trace, symmetric matrix, skew-symmetric matrix, and provides examples of solving systems of equations using matrices. 2. Examples include finding the trace of a matrix, determining if a matrix is symmetric or skew-symmetric, solving systems of equations using inverse matrices and Cramer's rule, and calculating determinants, adjoints and inverses. 3. Key steps shown are setting up matrices to represent systems of equations, taking the adjoint and inverse, and using the inverse matrix to solve the system of equations.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    14K views11 pages

    Matrices Inter First Year Important

    Uploaded by

    Syed Salman
    1. The document defines key terms related to matrices such as trace, symmetric matrix, skew-symmetric matrix, and provides examples of solving systems of equations using matrices. 2. Examples include finding the trace of a matrix, determining if a matrix is symmetric or skew-symmetric, solving systems of equations using inverse matrices and Cramer's rule, and calculating determinants, adjoints and inverses. 3. Key steps shown are setting up matrices to represent systems of equations, taking the adjoint and inverse, and using the inverse matrix to solve the system of equations.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 11
    At a glance
    Powered by AI
    The key takeaways are about different types of matrices like symmetric, skew-symmetric matrices and how to perform operations like finding the trace, inverse, adjoint and determinant of matrices. It also discusses how to solve systems of linear equations using matrix inversion and Cramer's rule.

    The different types of matrices discussed are symmetric, skew-symmetric and identity matrices. A symmetric matrix is equal to its transpose while a skew-symmetric matrix is equal to the negative of its transpose.

    To find the inverse of a matrix, you calculate the adjoint and divide it by the determinant. The adjoint of a matrix is the transpose of the cofactors. You can also use matrix inversion to solve systems of linear equations.

    Matrices

    1.

    VSAQ

    Sol:

    Define trace of the matrix and find trace of

    A+A=

    A=
    =

    AA=

    2.

    Sol: sum of elements in the principle diagonal of a


    square matrix is called trace of A.
    =1+ (-1) +1=1.
    Define a symmetric matrix, If
    A=

    =
    7.

    Sol:
    2X=B-A
    2X=

    Sol: A square matrix A is said to be symmetric matrix if


    A=A.

    3.

    If
    A=

    , then find x.

    x=6.

    X=
    8.

    Define skew symmetric,

    Find the cofactor of the elements 2, -5 in the


    matrix

    If
    Sol: the cofactor of the elements 2 is =

    Matrix, then find x.


    Sol: A square matrix A is said to be skew symmetric
    matrix

    the cofactor of the elements -5 is =


    =-(2-5)
    =-(-3)
    =3.

    if A=-A
    x=-4

    4.

    9.

    If A=

    Sol:

    Sol:
    1 (3x+24) =45
    3x=45-24
    3x=21 x=7

    5.

    Find the det of

    =-31-4(-44) +9(-17)
    =-31+176-153
    =176-184
    =-8.
    10. If A

    If
    Find the value of x, y, z and a.

    Sol: A

    Sol:

    =0

    Equating the corresponding elements


    x=5+3|| 2y=2+8 || z=-2-2 ||a=6+4
    x=8, y=5, z=-4 and a=10
    6.

    If

    -2-k=0k=-2.
    11. Find the ad joint and inverse of

    Matrices
    Sol:
    ifA=
    A=

    16. If A=
    =

    =
    A=

    12. If
    . (1)
    A+B=

    (2)

    From (1) & (2)

    =0.

    SAQ

    13. If A=
    Sol: A=

    1.

    If A=

    Sol: A=

    14. Find the rank of the matrix


    Sol:

    =
    Rank of A=3(number of non -zero rows)

    15. Construct a matrix A=


    .
    Let A=
    .

    2.

    If A=
    Sol:

    Matrices
    =

    5.

    If I=
    Sol:

    =
    =
    =

    =
    or

    =
    =

    3.

    If A=
    Sol:

    6.

    If A=
    Sol: let p (n) =

    = -4+1+4
    =1
    A

    Step-1: put n=1

    A=

    Step-2: let us assume that p(k) is true for n=k


    Put n=k
    Step-3: put n=k+1

    =
    4.

    Find the inverse of diag [a b c].


    Sol: let A=

    7.

    IfA=
    (h/w)
    Sol: s(n) is true for n=k+1.

    Matrices
    8.

    Show that

    Sol: L.H.S

    Sol:
    =

    =
    =

    =
    =
    12. If

    =
    =

    9.

    Show that

    Sol:

    , then show that

    Sol:

    (h/w)
    =cos

    10. Show that

    L.H.S

    =
    Sol: L.H.S
    =
    =
    =
    =

    =
    e

    11. Show that

    =-sin

    Matrices

    LAQ (2

    1.
    =
    Sol: L.H.S
    =

    (2)

    3.

    If

    Sol:

    R.H.S

    2.

    L.H.S

    Matrices
    =
    4.

    Expanding along

    Sol:

    6.

    S.T

    (a-b) (b-c) (c-a) (ab+bc+ca).

    Sol: L.H.S

    =
    =

    = (a-b) (b-c)

    Expanding along
    = (a-b) (b-c)

    =
    =
    b(c-a) +(c-a

    5.

    (c-a

    = (a-b) (b-c)
    Sol:
    = (a-b) (b-c)
    Expanding along
    = (a-b) (b-c)
    =

    = (a-b) (b-c)
    = (a-b) (b-c) (c-a) (ab+bc+ca).

    Matrices
    7.

    S.T

    (a-b) (b-c) (c-a)


    Sol: A=

    Sol:

    Now A. adjA=

    =
    =
    =
    =

    Expanding along
    =
    =
    (a-b) (b-c) (c-a)

    A.

    8.

    x=4.
    9.

    If A=

    =I

    Similarly we can prove that

    = detA.I

    =I

    Matrices
    10. Solve the following equations by using Cramers rule
    and matrix inversion method.
    a)
    b)

    x=3, y=1 and z=1.


    (b).
    Sol: let A=

    , X=

    and B=

    =2
    =2(1+1) +1(1-1) +3(-1-1)
    =2(2) +1(0) +3(-2)
    =4-6 = -2
    =6
    (a).
    Sol: let A=

    , X=

    and B=

    =9(1+1) +1(6-2) +3(-6-2)


    =9(2) +1(4) +3(-8)
    =18+4-24=22-24=-2
    =2

    =3

    -4

    +5

    =3(-7+16) -4(14-40) +5(-4+5)


    =3(9)-4(-26) +5(1)
    =27+10+5
    =136

    =18

    -4

    +5

    =18(-7+16) -4(91-160) +5(-26+20)


    =18(9)-4(-69) +5(-6)
    =162+276-30
    =408

    =2(6-2) -9(1-1) +3(2-6)


    =2(4) +1(0) +3(-24)
    =8-12=-4
    =2
    =2(2+6) +1(2-6) +9(-1-1)
    =2(8) +1(-4) +9(-2)
    =16-4-18
    =16-22=-6
    x=

    =1, y=

    , z=

    , X=

    and B=

    x=1, y=2 and z=3.


    (c). A=
    =1

    =3

    -18

    +5

    =3(91-160) -18(14-40) +5(40-65)


    =3(-69)-18(-26) +5(-25)
    =-207+468 -125
    =136

    =3

    -4

    +18

    =3(-20+26) -4(40-65) +5(-4+5)


    =3(6)-4(-25)+5(1)
    =18+100+18
    =136
    x=

    , y=

    , z=

    =1(-5-7) -1(-2-14) +1(2-10)


    =1(-12) -1(-16) +1(-8)
    =-12+16-8=- 4
    =9
    =9(-5-7) -1(-52-0) +1(52-0)
    =9(-12) -1(-52) +1(52)
    =-108+52+52
    =-108+104=-4
    =

    =1

    =1(-52-0) -9(-2-14) +1(0-104)


    =1(-52) -9(-16) +1(-104)

    Matrices
    =-52+144-104
    =-156+144=-12

    AdjA=
    =1

    =1(0-52) -1(0-104) +9(2-10)


    =1(-52) -1(-104) +9(-8)
    =-52+104-72
    =-124+104=-20
    x=

    =1, y=

    , z=

    x=1, y=3 and z=5.


    (d). A=

    , X=

    and B=

    AdjA =

    =2
    DetA=
    2(9)-4(-26) +5(1)
    =136

    =2(-8-1) +1(4-3) +3(-1-6)


    =2(-9) +1(1) +3(-7)
    =--18+1-21
    =-38

    A-1=
    X=A-1.B

    =8

    =8(-8-1) +1(-16-0) +3(4-0)


    =8(-9) +1(-16) +3(4)
    =-72-16+12
    =-76

    =2
    ==

    =2(-16-0) -8(4-3) +3(0-12)


    =2(-16) -8(1) +3(-12)
    =-32-8-36
    =-76

    (b). Sol: let A=

    And A-1=

    =2(0-4+1(0-12+8(-1-6)
    =2(-4) +1(-12) +8(-7)
    =-8-12-56
    =76
    =2, y=

    =2

    , z=

    x=2, y=2 and z=2.


    Matrix inversion method:
    (a). A=
    AX=BX=A-1.B
    And A-1=

    , X=

    AX=BX=A-1.B

    =2

    x=

    x=3, y=1 and z=1.

    , X=

    =2(1+1) +1(1-1) +3(-1-1)


    =2(2) +1(0) +3(-2)
    =4-6 = -2
    AdjA=

    and B=
    =

    and B=

    Matrices
    =
    =
    ==

    x=1, y=3 and z=5.

    AdjA =
    (d).let A=

    , X=

    and B=

    A-1=
    =2

    X=A-1.B

    =2(-8-1) +1(4-3) +3(-1-6)


    =2(-9) +1(1) +3(-7)
    =--18+1-21
    =-38

    AdjA=
    ==
    (c). A=

    x=1, y=2 and z=3


    , X=
    =1

    =1(-5-7) -1(-2-14) +1(2-10)


    =1(-12) -1(-16) +1(-8)
    =-12+16-8=- 4
    AdjA=

    and B=

    AdjA =
    A-1=
    X=A-1.B

    AdjA =

    ==

    A-1=
    X=A-1.B
    =

    x=2, y=2 and z=2.

    Matrices

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