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    MATRICES

    WELCOME

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    M TRIX A rectangular arrangement ofnumbers in rows and columns.

    The OR ERof a matrix is the number of therows and columns.

    The ENTRIESare the numbers in the matrix.

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    67237

    89511

    36402

    0759

    3410

    200318

    20

    11

    6

    0

    7

    9

    3 x 3

    3 x 5

    2 x 2 4 x 1

    1 x 4

    (or squarematrix)

    (Also called a rowmatrix)

    (or squarematrix)

    (Also called acolumn matrix)

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    IfAand Bare both mn matrices then the sumofAand B, denotedA+ B, is amatrix obtained by adding correspondingelements ofAand B.

    310

    221A

    412

    403B

    102

    622BA

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    To subtract two matrices, they must have the sameorder. You simply subtract corresponding entries.

    232

    451

    704

    831

    605

    429

    2833)2(1

    )4(65015740249

    603

    1054

    325

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    In matrix algebra, a real number is often called a SCALAR. Tomultiply a matrix by a scalar, you multiply each entry in thematrix by that scalar.

    )1(4)4(4

    )0(4)2(4

    14

    024

    416

    08

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    ABBA

    CBACBA )()(

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    IfAis an mn matrix and sis a scalar, then we let kA denote the matrix obtainedby multiplying every element ofA by k. This procedure is called scalar

    multiplication.

    k hA kh A

    k h A kA hA

    k A B kA kB

    310

    221A

    930

    663

    331303

    2323133A

    PROPERTIES OF SCALAR MULTIPLICATION

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    The mnzero matrix, denoted 0, is the mn

    matrix whose elements are all zeros.

    000)(

    0

    A

    AA

    AA

    00

    00 000

    2 21 3

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    BCACCBAACABCBA

    CABBCA

    PROPERTIES OF MATRIX

    MULTIPLICATION

    BAAB

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    an nnmatrix with ones on the main diagonal and zeros

    elsewhere

    100

    010

    001

    3

    I

    What isAI?

    What is IA?

    322

    510

    212

    A

    100

    010

    001

    3I

    A

    322

    510

    212

    A

    322

    510

    212

    Multiplying a matrix by

    the identity gives the

    matrix back again.

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    IfAhas an inverse we say thatAis nonsingular.

    IfA-1does not exist we sayAis singular.

    To find the inverse of a matrix we put the matrix A, a line and then the

    identity matrix. We then perform row operations on matrix A to turn it into

    the identity. We carry the row operations across and the right hand side

    will turn into the inverse.

    To find the inverse of a matrix we put the matrix A, a line and then the

    identity matrix. We then perform row operations on matrix A to turn it into

    the identity. We carry the row operations across and the right hand side

    will turn into the inverse.

    72

    31A

    1210

    0131

    2r

    1+r

    2

    1072

    0131

    12100131r2

    1210

    3701r1 r2

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    bx 1 A

    bx 11 AAA

    bxA left multiply both sides by theinverse of A

    This is just the identity

    bx 1 AI

    but the identity times a matrix

    just gives us back the matrix

    so we have:

    This then gives us a formula for

    finding the variable matrix: MultiplyA

    inverse by the constants.

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    Remember You cant multiply anymatrix. There are some conditionsThe number of columns on the first matrix has to be

    same as the number of rows of the second matrix.

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    The Transpose of a Matrix Sometimes it is of interest to interchange the rows

    and columns of a matrix

    The transpose of a matrix A=Aijis a matrix formedfrom A by inter changing rows and columns suchthat row i of A becomes columns I of the transposematrix. The transpose is denoted by At and

    At=Aji when A= Aij

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    Example of the Transpose A= 1 3 AT= 1 2

    2 5 3 5

    A= 1 3 4 AT= 1 00 1 0 3 1

    4 0

    It will be observed that ifAis m x n,At isn x m

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    Q. What is a Part i t ioned Matr ixand what

    does it have to do with me?

    A. Ah, good question.

    Well, a Part i t ioned Matr ixis a matrix that has

    been broken down into several smaller matrices

    But why tell you when I can show you a picture.

    Lets say I have a 5x4 Matrix called G

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    And now a partitioned version (with the partitionlines in red):

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    And now we name the individual parts

    (AKA: Blocksor Submatr ices):

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    Now we can rewrite G as a 3x2

    Matrix:

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    MIND READER

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    1.You should press thecalculator thrice

    You should the result as 100The only condition is that you

    should not press 0 button

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    2. A 40kg stone is broken into 4pieces.

    What will be the wait of 4

    broken pieces.You have to tell the wait of 4

    pieces provided that the values

    must satisfies every kg upto 40 kg

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    V.VijayII M.sc. Mathematics120814