Key Words

Topics: Multiplication, Matrix, Addition Pages: 35 (1459 words) Published: September 25, 2014
3.1
Matrices
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Matrices
Matrices are classified in terms of the numbers of rows and
columns they have. Matrix M has three rows and four
columns, so we say this is a 3  4 (read “three by four”) matrix.

2

Matrices
The matrix

has m rows and n columns, so it is an m  n matrix. When
we designate A as an m  n matrix, we are indicating the
size of the matrix.

3

Matrices
Two matrices are said to have the same order (be the
same size) if they have the same number of rows and the
same number of columns.
For example,

and

do not have the same order.
C is a 2  3 matrix and D is a 3  2 matrix.
4

Matrices
The numbers in a matrix are called its entries or elements.
Note that the subscripts on an entry in matrix A above
correspond respectively to the row and column in which the
entry is located.
Thus a23 represents the entry in the second row and the
third column, and we refer to it as the “two-three entry.” In matrix B below, the entry denoted by b23 is 1.
Some matrices take special names because of their size. If
the number of rows equals the number of columns, we say
the matrix is a square matrix.
5

Matrices
Matrix B below is a 3  3 square matrix.

A matrix with one row, such as [9 5] or [3 2 1 6], is called a row matrix, and a matrix with one column, such as

is called a column matrix.
6

Matrices
Row and column matrices are also called vectors.

Any matrix in which every entry is zero is called a zero
matrix; examples include

and

We define two matrices to be equal if they are of the same
order and if each entry in one equals the corresponding
entry in the other.
7

Matrices
When the columns and rows of matrix A are interchanged
to create a matrix B, and vice versa, we say that A and B
are transposes of each other and write AT = B and BT = A.

8

Example 2 – Matrices
(a) Which element of

is represented by a32?
(b) Is A a square matrix?

(c) Find the transpose of matrix A.
(d) Does A = B?
9

Example 2 – Solution

10

Addition and Subtraction of Matrices
Addition and Subtraction
of Matrices
If two matrices have the same number of rows and
columns, we can add the matrices by adding their
corresponding entries.

11

Addition and Subtraction of Matrices
Sum of Two Matrices

12

Addition and Subtraction of Matrices
Note that the sum of A and B could be found by adding the
matrices in either order. That is, A + B = B + A.
This is known as the commutative law of addition for
matrices.

The matrix –B is called the negative of the matrix B, and
each element of –B is the negative of the corresponding
element of B.

13

Addition and Subtraction of Matrices
For example, if

then

Using the negative, we can define the difference A – B
(when A and B have the same order) by A – B = A + (–B),
or by subtracting corresponding elements.

14

Example 5 – Balance of Trade
Table 3.2 summarizes the dollar value (in millions) of 2006
U.S. exports and imports of cars, trucks, and automotive
parts for selected countries.

Table 3.2

Write the matrix that describes the balance of trade with the selected countries for cars, trucks, and parts.
15

Example 5 – Solution
From Table 3.2 we represent the exports as matrix A and
the imports as matrix B.

The balance of trade we seek is given by the difference
A – B, as follows.

16

Example 5 – Solution

cont’d

Note that any negative entry indicates an unfavorable
balance of trade for the item and with the country
corresponding to that entry.
Also note that the 1–3 entry is the only positive one and
that it indicates a favorable trade balance, in automotive
parts with Canada, worth $11,820,000,000.

17

Scalar Multiplication
Scalar Multiplication
We can define scalar multiplication as follows.

Scalar Multiplication

18

Example 6 – Scalar Multiplication
If

find 5A and...
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