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Engineering LibreTexts

2.10: Untitled Page 22

  • Page ID
    18155
  • Chapter 2

    3 x  4 y  2 z   2

    8 x  2 y  7 z

    5

    (2‐38)

    5 x  3 y  4 z

    3

    In matrix form, we express this system of equations according to

     3

    4

    2   x

     2

     

      

    8 2 7

    y

    5

      

    (2‐39)

     5 

    3

    4  

      z 

     3 

    The rule for multiplication implied here is that the first row of the 3  3 matrix multiplies the 3  1 unknown column matrix to obtain the first element of the 3  1

    known column matrix. In terms of compact notation, we use Arial font to represent matrices and this leads to the compact form of Eq. 2‐39 given by Au

     b

    (2‐40)

    Here A represents the 3  3 matrix in Eq. 2‐39

     3

    4

    2 

    A

    8 2 7

    (2‐41)

     5 

    3

    4 

    while u and b represent the two 3  1 column matrices according to

    x

     2 

     

    u

    y

    ,

    b

    5

     

    (2‐42)

    z

     3 

     

    The 3  1 matrices are sometimes called column vectors; however, the word vector should be reserved for a quantity that has magnitude and a direction, such as a force, a velocity, or an acceleration. In this text, we will use the phrase column matrix for a n  1 matrix and row matrix for a 1 n matrix. Examples of a 1 4

    row matrix and a 4  1 column matrix are given by

     6 

    2

    b

      3 1 5 6  ,

    c

     

    (2‐43)

    0 

     4 

    Units

    31

    In a matrix, the numbers are ordered in a rectangular grid of rows and columns, and we indicate the size of an array by the number of rows and columns. The following is a 4  4 matrix denoted by A :

     3

    1

    5

    6 

    4

    3

    6

    2

    A

     

    (2‐44)

    8

    3

    2

    0 

     1

    5

    8

    4 

    Matrices have a well‐defined algebra that we will explore in more detail in subsequent chapters. At this point we will introduce only the operations of scalar multiplication, addition and subtraction. Scalar multiplication of a matrix consists of multiplying each element of the matrix by a scalar, thus if c is any real number, the scalar multiple of Eq. 2‐44 is given by

    3 c

    1 c

    5 c

    6 c

    4 c 3 c

    6 c

    2

    c

    c A  

    (2‐45)

    8 c

    3 c

    2 c

    0 

    1 c

    5 c

    8 c

    4 c

    Matrices have the same size when they have the same number of rows and columns. For example, the two matrices A and B

     11

    a

    12

    a

    ......

    1

    a n

     11

    b

    12

    b

    ......

    1

    b n

    a

    a

    ...... a

    b

    b

    ...... b

    21

    22

    2 n

    21

    22

    2

    A

     

     ,

    n

    B

     

    (2‐46)

    ....

    ....

    ......

    .... 

     ....

    ....

    ......

    .... 

    a 1

    m

    m

    a 2 ......

    m

    a n

    b 1

    m

    m

    b 2 ......

    m

    b n

    have the same size and the sum of A and B is created by adding the corresponding elements to obtain

    11

    a

    11

    b

    12

    a

    12

    b

    ......

    1

    a n

    1

    b n

    a b

    a

    b

    ......

    a

    b

    21

    21

    22

    22

    2 n

    2 n

    A  B

     

    (2‐47)

    ....

    ....

    ......

    ....

    a

     1

    m

    b 1

    m

    m

    a 2

    m

    b 2 ......

    mn

    a

    mn

    b

    It should be apparent that the sum of two matrices of different size is not defined, and that subtraction is carried out in the obvious manner indicated by

    32