Matrices Determinants MS

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    MATRICES &DETERMINANTS

    Monika V SikandLight and Life Laboratory

    Department of Physics and Engineering physicsStevens Institute of TechnologyHoboken, New Jersey, 07030.

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    OUTLINE

    Matrix Operations

    Multiplying MatricesDeterminants and Cramers Rule Identity and Inverse Matrices

    Solving systems using Inverse matrices

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    MATRIX

    A rectangular arrangement of numbers in rows and columns

    For example:

    6 2 12 0 5

    2 rows

    3 columns

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    TYPES OF MATRICES

    NAME DESCRIPTION EXAMPLE

    Row matrix A matrix with only 1row

    Column matrix A matrix with only Icolumn

    Square matrix A matrix with samenumber of rows andcolumns

    Zero matrix A matrix with all zeroentries

    3 2 1 4 2

    3

    2 4

    1 7

    0 0

    0 0

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    MATRIX OPERATIONS

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    COMPARING MATRICES

    For Example:

    5 044

    34

    5 0

    1 0.75

    2 6

    0 3

    2 6

    3 2

    EQUAL MATRICES: Matrices having equal correspondingentries.

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    ADDING MATRICES

    Matrices of same dimension can be added

    For Example:

    3

    4

    2

    1

    0

    2

    3 1

    4 0

    2 2

    4

    4

    5

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    SUBTRACTING MATRICES

    Matrices of same dimension can be subtracted

    For example:

    8 3

    4 0

    2 7

    6 1

    8 2 3 ( 7)

    4 6 0 ( 1)

    6 10

    2 1

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    MULTIPLYING A MATRIX BY ASCALAR

    For example:

    2

    1 2

    0 34 5

    4 5

    6 82 6

    ( 2)1 ( 2) 2

    ( 2)0 ( 2)3( 2) 4 ( 2)5

    4 5

    6 82 6

    2 4

    0 6

    8 10

    4 5

    6 8

    2 6

    6 9

    6 14

    6 4

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    SOLVING A MATRIXEQUATION

    For example: Solve :

    23 x 1

    8 5

    4 1

    2 y

    26 0

    12 8

    23 x 4 1 1

    8 2 5 y

    26 0

    12 8

    6 x 8 0

    12 10 2 y

    26 0

    12 8

    Equate :

    6 x 8 26

    x 3

    10 2 y 8

    y 1

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    MULTIPLYING MATRICES

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    PRODUCT OF TWO MATRICES

    A 3 21 0

    B1 4

    2 1

    For example:

    FIND (a.) AB and (b.) BA

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    SOLUTION

    AB3 2

    1 0

    1 4

    2 1

    AB7 10

    1 4

    BA1 4

    2 1

    3 2

    1 0

    BA7 2

    5 4

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    SIMPLIFY

    A2 1

    1 3, B

    2 0

    4 2,C

    1 1

    3 2Simplify:a.) A(B+C)

    b.) AB+AC

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    SOLUTION

    2 1

    1 3

    2 1

    1 3

    1 1

    3 2 2 1

    1 3

    1 1

    7 4

    5 6

    22 11

    A(B+C):

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    SOLUTION

    AB+AC:

    2 1

    1 3

    2 0

    4 2

    2 1

    1 3

    1 1

    3 2

    0 2

    14 6

    5 4

    8 5

    5 6

    22 11

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    DETERMINANTS &CRAMERS RULE

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    DETERMINANT OF 2 2 MATRIX

    deta b

    c d ad bc

    The determinant of a 2 2 matrix is the difference of the entries on the diagonal.

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    EVALUATE

    Find the determinant of the matrix:

    1 3

    2 5Solution:

    1 32 5

    1(5) 2(3) 5 6 1

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    DETERMINANT OF 3 3 MATRIX

    The determinant of a 3 3 matrix is the difference in thesum of the products in red from the sum of the productsin black.

    det

    a b c

    d e f

    g h i

    a b c

    d e f

    g h i

    a b

    d e

    g h

    Determinant = [a(ei)+b(fg)+c(dh)]- [g(ec)+h(fa)+i(db)]

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    EVALUATE

    2 1 3

    2 0 1

    1 2 4

    2 1 3

    2 0 1

    1 2 4

    2 1

    2 0

    1 2

    [0 ( 1) ( 12)] (0 4 8) 13 12 25

    Solution:

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    USING MATRICES IN REAL LIFE

    The Bermuda Triangle is a large trianglular region inthe Atlantic ocean. Many ships and airplanes havebeen lost in this region. The triangle is formed by

    imaginary lines connecting Bermuda, Puerto Rico, andMiami, Florida. Use a determinant to estimate the areaof the Bermuda Triangle.

    EW

    N

    S

    Miami (0,0)

    Bermuda (938,454)

    Puerto Rico (900,-518)

    ...

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    SOLUTION

    The approximate coordinates of the Bermuda Triangles three vertices are: (938,454), (900,-518), and (0,0). Sothe area of the region is as follows:

    Area 12

    938 454 1

    900 518 1

    0 0 1

    Area 12

    [( 458,884 0 0) (0 0 408,600)]

    Area 447,242

    Hence, area of the Bermuda Triangle is about 447,000square miles.

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    USING MATRICES IN REAL LIFE

    The Golden Triangle is a large triangular region in theIndia.The Taj Mahal is one of the many wonders that liewithin the boundaries of this triangle. The triangle isformed by the imaginary lines that connect the cities of New Delhi, Jaipur, and Agra. Use a determinant toestimate the area of the Golden Triangle. Thecoordinates given are measured in miles.

    EW

    N

    S

    Jaipur (0,0)

    New Delhi (100,120)

    Agra (140,20)

    . ..

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    SOLUTION

    The approximate coordinates of the Golden Triangles three vertices are: (100,120), (140,20), and (0,0). So thearea of the region is as follows:

    Area 12

    100 120 1

    140 20 1

    0 0 1

    Area 12

    [(2000 0 0) (0 0 16800)]

    Area 7400

    Hence, area of the Golden Triangle is about 7400 squaremiles.

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    USING MATRICES IN REAL LIFE

    Black neck stilts are birds that live throughout Floridaand surrounding areas but breed mostly in thetriangular region shown on the map. Use a determinant to estimate the area of this region. The coordinatesgiven are measured in miles.

    EW

    N

    S

    (0,0)

    (35,220)

    (112,56)

    . .

    .

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    SOLUTION

    The approximate coordinates of the Golden Triangles three vertices are: (35,220), (112,56), and (0,0). So thearea of the region is as follows:

    Area 12

    35 220 1

    112 56 1

    0 0 1

    Area 12

    [(1960 0 0) (0 0 24640)]

    Area 11340

    Hence, area of the region is about 11340 square miles.

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    CRAMERS RULE FOR A 2 2SYSTEM

    Let A be the co-efficient matrix of the linear system:ax+by= e & cx+dy= f.

    IF det A 0, then the system has exactly one solution.The solution is:

    x

    e b

    f d det A

    y

    a e

    c f

    det A

    The numerators for x and y are the

    determinant of the matrices formed byusing the column of constants asreplacements for the coefficients of x andy, respectively.

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    EXAMPLE

    Use cramers rule to solve this system:

    8x+5y = 22x-4y = -10

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    SOLUTION

    Solution: Evaluate the determinant of the coefficient matrix

    8 5

    2 4 32 10 42

    Apply cramers rule since the determinant is not zero.

    x

    2 510 4

    428 ( 50)

    424242

    1

    y

    8 2

    2 1042

    80 442

    8442 2 The solution is (-1,2)

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    CRAMERS RULE FOR A 3 3SYSTEM

    Let A be the co-efficient matrix of the linear system:ax+by+cz= j, dx+ey+fz= k, and gx+hy+iz=l.

    IF det A 0, then the system has exactly one solution.The solution is:

    x

    j b c

    k e f

    l h i

    det A

    , y

    a j c

    d k f

    g l i

    det A

    , z

    a b j

    d e k

    g h l

    det A

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    EXAMPLE

    The atomic weights of three compounds are shown. Use alinear system and Cramers rule to find the atomic weightsof carbon(C ), hydrogen(H), and oxygen(O).

    Compound Formula Atomic weight

    Methane CH 4 16

    Glycerol C 3H8O 3 92

    Water H 2O 18

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    SOLUTION

    1 4 0

    3 8 3

    0 2 1

    (8 0 0) (0 6 12) 10

    Write a linear system using the formula for each compound

    C + 4H = 16

    3C+ 8H + 3O = 922H + O =18

    Evaluate the determinant of the coefficient matrix .

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    SOLUTION

    Apply cramers rule since determinant is not zero.

    C

    16 4 0

    92 8 3

    18 2 110

    12010

    12

    H

    1 16 0

    3 92 3

    0 18 110

    1010

    1

    O

    1 4 16

    3 8 92

    0 2 1810

    16010

    16

    Atomic weight of carbon = 12

    Atomic weight of hydrogen =1

    Atomic weight of oxygen =16

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    IDENTITY AND INVERSEMATRICES

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    IDENTITY MATIX

    2 2 IDENTITY MATRIX 3 3 IDENTITY MATRIX

    I 1 0

    0 1 I

    1 0 0

    0 1 0

    0 0 1

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    INVERSE MATRIX

    The inverse of the matrix

    A

    a b

    c d

    is

    A 11 A

    d b

    c a

    A 11

    ad cb

    d b

    c a

    provided

    ad cb 0

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    EXAMPLE

    Find the inverse of

    A3 1

    4 2

    Solution:

    A 11

    6 4

    2 1

    4 312

    2 1

    4 3

    11

    2

    232

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    CHECK THE SOLUTION

    Show

    AA 1 I A 1 A

    3 1

    4 2

    1 12

    232

    1 0

    0 1,

    and

    112

    232

    3 1

    4 2

    1 0

    0 1

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    SOLVING SYSTEMS USING

    INVERSE MATRICES

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    SOLVING A LINEAR SYSTEM

    -3x + 4y = 52x - y = -10

    Writing the original matrix equation.

    3 4

    2 1

    x

    y

    5

    10

    A X B AX = B A -1 AX = A -1B

    IX = A-1

    BX = A -1B

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    USING INVERSE MATRIX TOSOLVE THE LINEAR SYSTEM

    -3x + 4y = 52x - y = -10

    A 11

    3 8

    1 4

    2 3

    15

    45

    25

    35

    X A 1 B

    1

    5

    4

    525

    35

    510

    74

    x y

    Hence the solution of the system is (-7,-4)