# Key Words

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?

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Example 2 – Solution

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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.

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Addition and Subtraction of Matrices

Sum of Two Matrices

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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.

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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.

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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.

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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.

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Scalar Multiplication

Scalar Multiplication

We can define scalar multiplication as follows.

Scalar Multiplication

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Example 6 – Scalar Multiplication

If

find 5A and...

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