606 Chapter 18 Mathematics in Engineering
by a single value such as two hours. These types of physical variables which are identifiable by a
single value are calledscalars. Temperature is another example of a scalar variable. On the other
hand, if you were to describe the velocity of a vehicle, you not only have to specify how fast it is
moving (speed), but also its direction. The physical variables that possess both magnitude and
direction are calledvectors. There are also other quantities that require specifying more than two
pieces of information to describe them accurately. For example, if you were to describe the loca-
tion of a car parked in a multi-story garage (with respect to the garage entrance), you would need
to specify the floor (zcoordinate), and then the location of the car on that floor (xandycoordi-
nates). A matrix is often used to describe situations that require many values. Amatrixis an array
of numbers, variables, or mathematical terms. The numbers or the variables that make up the matrix
are called theelements of matrix. Thesizeof a matrix is defined by its number of rows and columns.
A matrix may consist ofmrows andncolumns. For example,matrix [N] is a 3 by 3 (or 3 3) matrix whose elements are numbers, and {L} is a 3 by 1
matrix with its elements representing variablesx,y, andz. The [N] is called a square matrix. A
squarematrix has the same number of rows and columns. The element of a matrix is denoted
by its location. For example, the element in the first row and the third column of a matrix [N]
is denoted byn 13 (it readsnsub 13), which has a value of 9. In this book, we denote the matrix
by a boldface letterin brackets like [ ] and { }. For example, [N], [T], {F}, and the elements
of matrices are represented by regular lowercase letters. The { } is used to distinguish a column
matrix. A column matrix is defined as a matrix that has one column but could have many rows.
On the other hand, a row matrix is a matrix that has one row but could have many columns.
Examples of column and row matrices follow.are examples of column matrices, whereas
are examples of row matrices.
Diagonal and Unit Matrices A diagonal matrix is one that only has elements along its principal
diagonal; the elements are zero everywhere else. An example of a 4 4 diagonal matrix
follows.3 A 4 ≥
500 0
070 0
004 0
00011
¥
3 C 4 3502 34 and 3 Y 4 3 y 1 y 2 y 34
5 A 6 μ
1
5
2
3
∂ and 5 X 6 •
x 1
x 2
x 3
¶
5 L 6 •
x
y
z
3 N 4 £ ¶
659
12614
580
§
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