Machinery's Handbook, 31st Edition
MATRICES
123
MATRICES Multiple variables are often present in technology and engineering scenarios; some ex- amples include electrical circuitry, in which there is a long series of resistors, and cost analysis (labor, materials, capital) in industrial economics. A system of linear equations can be solved by several methods, such as substitution and elimination (see pages 32 to 33, Solving a System of Linear Equations in ALGE- BRA ). Another way to solve a system, useful especially when more than two variables are involved, is to set up a matrix of the equations’ coefficients. A matrix consists of real numbers arranged in horizontal rows and vertical columns to form a rectangular array. An array of m rows and n columns is an m × n matrix (read as “ m by n ”) and is written as a 11 … 0 . . .
. .
.
. . . .
.. .
.. .
. .
.
. .
a
… . . . The a ij terms are called the entries or elements of the matrix. The subscript i identi fies the row position of an entry, and the subscript j identifies its column position in the matrix. For example, in the matrix below, a 11 = 3, a 12 = 4, a 21 = –1, and a 22 = 2: mn 0
3 4 –1 2 Special matrices used in matrix operations include:
a 1
a
a 11 a m 1 . . .
… . . .
0 … 0 . . . . . . . . . . . . . . . . . .
1 n
. . . . . . . . .
. .
. . . .
a 1 n
a ( . . . 11
)
.
. .
.
. .
a
a
… . . .
0 … 0 . . .
n 1
nn
Column Matrix
Row Matrix (1 × n )
Square Matrix
Zero, or Null, Matrix ( n × n )
( m × 1)
( n × n )
The two types of special square matrices include:
a 1
a 1 n
0
… . . .
1 … . . . . . . .
. . .
. .
. . . . . . . . .
. . . . . . . .
. . . .
.
. .
.
. .
a
… 1 a n 1 . . .
0
… . . .
nn
Diagonal Matrix ( n × n ) All entries are 0, except possibly those on the diagonal.
Identity Matrix ( n × n ) All diagonal entries are 1, all others are 0.
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