Matrices: Basic Operations

Содержание

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Barnett/Ziegler/Byleen Finite Mathematics 11e Addition and Subtraction of Matrices To add

Barnett/Ziegler/Byleen Finite Mathematics 11e

Addition and Subtraction of Matrices

To add or

subtract matrices, they must be of the same order, m x n. To add matrices of the same order, add their corresponding entries. To subtract matrices of the same order, subtract their corresponding entries. The general rule is as follows using mathematical notation:
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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Addition Add the matrices

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Addition

Add the matrices

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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Addition Solution Add the matrices Solution:

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Addition Solution

Add the matrices
Solution: First note that

each matrix has dimensions of 3x3, so we are able to perform the addition. The result is shown at right:

Adding corresponding entries, we have

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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Subtraction Now, we will subtract the same two matrices =

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Subtraction

Now, we will subtract the same two

matrices

=

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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Subtraction Solution Now, we will subtract

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Subtraction Solution

Now, we will subtract the same two

matrices

Subtract corresponding entries as follows:

=

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Barnett/Ziegler/Byleen Finite Mathematics 11e Scalar Multiplication The scalar product of a

Barnett/Ziegler/Byleen Finite Mathematics 11e

Scalar Multiplication

The scalar product of a number k

and a matrix A is the matrix denoted by kA, obtained by multiplying each entry of A by the number k. The number k is called a scalar. In mathematical notation,
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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Scalar Multiplication Find (-1)A, where A =

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Scalar Multiplication

Find (-1)A, where A =

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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Scalar Multiplication Solution Find (-1)A, where A = Solution: (-1)A= -1

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Scalar Multiplication Solution

Find (-1)A, where A =

Solution:
(-1)A=

-1
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Barnett/Ziegler/Byleen Finite Mathematics 11e Alternate Definition of Subtraction of Matrices The

Barnett/Ziegler/Byleen Finite Mathematics 11e

Alternate Definition of Subtraction of Matrices

The definition

of subtraction of two real numbers a and b is a – b = a + (-1)b or “a plus the opposite of b”. We can define subtraction of matrices similarly:

If A and B are two matrices of the same dimensions, then A – B = A + (-1)B, where (-1) is a scalar.

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Barnett/Ziegler/Byleen Finite Mathematics 11e Example The example on the right illustrates

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example

The example on the right illustrates this procedure

for two 2x2 matrices.
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Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Equations Example: Find a, b, c, and d so that

Barnett/Ziegler/Byleen Finite Mathematics 11e

Matrix Equations

Example: Find a, b, c, and d

so that
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Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Equations Example: Find a, b, c,

Barnett/Ziegler/Byleen Finite Mathematics 11e

Matrix Equations

Example: Find a, b, c, and d

so that

Solution: Subtract the matrices on the left side:

Use the definition of equality to change this matrix equation into 4 real number equations:
a - 2 = 4 b + 1 = 3 c + 5 = -2 d - 6 = 4 a = 6 b = 2 c = -7 d = 10

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Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Products The method of multiplication of

Barnett/Ziegler/Byleen Finite Mathematics 11e

Matrix Products

The method of multiplication of matrices

is not as intuitive and may seem strange, although this method is extremely useful in many mathematical applications.

Matrix multiplication was introduced by an English mathematician named Arthur Cayley (1821-1895). We will see shortly how matrix multiplication can be used to solve systems of linear equations.

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Barnett/Ziegler/Byleen Finite Mathematics 11e Arthur Cayley 1821-1895 Introduced matrix multiplication

Barnett/Ziegler/Byleen Finite Mathematics 11e

Arthur Cayley 1821-1895

Introduced matrix multiplication

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Barnett/Ziegler/Byleen Finite Mathematics 11e Product of a Row Matrix and a

Barnett/Ziegler/Byleen Finite Mathematics 11e

Product of a Row Matrix and a Column

Matrix

In order to understand the general procedure of matrix multiplication, we will introduce the concept of the product of a row matrix by a column matrix.
A row matrix consists of a single row of numbers, while a column matrix consists of a single column of numbers. If the number of columns of a row matrix equals the number of rows of a column matrix, the product of a row matrix and column matrix is defined. Otherwise, the product is not defined.

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Barnett/Ziegler/Byleen Finite Mathematics 11e Row by Column Multiplication Example: A row

Barnett/Ziegler/Byleen Finite Mathematics 11e

Row by Column Multiplication

Example: A row matrix consists

of 1 row of 4 numbers so this matrix has four columns. It has dimensions 1 x 4. This matrix can be multiplied by a column matrix consisting of 4 numbers in a single column (this matrix has dimensions 4 x 1).
1x4 row matrix multiplied by a 4x1 column matrix. Notice the manner in which corresponding entries of each matrix are multiplied:
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Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Revenue of a Car Dealer A

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: Revenue of a Car Dealer

A car dealer sells

four model types: A, B, C, D. In a given week, this dealer sold 10 cars of model A, 5 of model B, 8 of model C and 3 of model D. The selling prices of each automobile are respectively $12,500, $11,800, $15,900 and $25,300. Represent the data using matrices and use matrix multiplication to find the total revenue.
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Barnett/Ziegler/Byleen Finite Mathematics 11e Solution using Matrix Multiplication We represent the

Barnett/Ziegler/Byleen Finite Mathematics 11e

Solution using Matrix Multiplication

We represent the number of

each model sold using a row matrix (4x1), and we use a 1x4 column matrix to represent the sales price of each model. When a 4x1 matrix is multiplied by a 1x4 matrix, the result is a 1x1 matrix containing a single number.
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Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Product If A is an m

Barnett/Ziegler/Byleen Finite Mathematics 11e

Matrix Product

If A is an m x

p matrix and B is a p x n matrix, the matrix product of A and B, denoted by AB, is an m x n matrix whose element in the i th row and j th column is the real number obtained from the product of the i th row of A and the j th column of B. If the number of columns of A does not equal the number of rows of B, the matrix product AB is not defined.
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Barnett/Ziegler/Byleen Finite Mathematics 11e Multiplying a 2x4 matrix by a 4x3

Barnett/Ziegler/Byleen Finite Mathematics 11e

Multiplying a 2x4 matrix by a 4x3 matrix

to obtain a 2x3

The following is an illustration of the product of a 2x4 matrix with a 4x3. First, the number of columns of the matrix on the left must equal the number of rows of the matrix on the right, so matrix multiplication is defined. A row-by column multiplication is performed three times to obtain the first row of the product: 70 80 90.

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Barnett/Ziegler/Byleen Finite Mathematics 11e Final Result

Barnett/Ziegler/Byleen Finite Mathematics 11e

Final Result

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Barnett/Ziegler/Byleen Finite Mathematics 11e Undefined Matrix Multiplication Why is the matrix multiplication below not defined?

Barnett/Ziegler/Byleen Finite Mathematics 11e

Undefined Matrix Multiplication

Why is the matrix multiplication below

not defined?
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Barnett/Ziegler/Byleen Finite Mathematics 11e Undefined Matrix Multiplication Solution Why is the

Barnett/Ziegler/Byleen Finite Mathematics 11e

Undefined Matrix Multiplication Solution

Why is the matrix multiplication below

not defined? The answer is that the left matrix has three columns but the matrix on the right has only two rows. To multiply the second row [4 5 6] by the third column, 3 , there is no number to pair with 6 to multiply. 7
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Barnett/Ziegler/Byleen Finite Mathematics 11e Example Given A = B = Find AB if it is defined:

Barnett/Ziegler/Byleen Finite Mathematics 11e

Example

Given A = B =
Find AB if it is

defined:
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Barnett/Ziegler/Byleen Finite Mathematics 11e Solution Since A is a 2 x

Barnett/Ziegler/Byleen Finite Mathematics 11e

Solution

Since A is a 2 x 3

matrix and B is a 3 x 2 matrix, AB will be a 2 x 2 matrix.
1. Multiply first row of A by first column of B: 3(1) + 1(3) +(-1)(-2)=8
2. First row of A times second column of B: 3(6)+1(-5)+ (-1)(4)= 9
3. Proceeding as above the final result is

=

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Barnett/Ziegler/Byleen Finite Mathematics 11e Is Matrix Multiplication Commutative? Now we will

Barnett/Ziegler/Byleen Finite Mathematics 11e

Is Matrix Multiplication Commutative?

Now we will attempt

to multiply the matrices in reverse order: BA = ?
We are multiplying a 3 x 2 matrix by a 2 x 3 matrix. This matrix multiplication is defined, but the result will be a 3 x 3 matrix. Since AB does not equal BA, matrix multiplication is not commutative.

=

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Barnett/Ziegler/Byleen Finite Mathematics 11e Practical Application Suppose you a business owner

Barnett/Ziegler/Byleen Finite Mathematics 11e

Practical Application

Suppose you a business owner and sell

clothing. The following represents the number of items sold and the cost for each item. Use matrix operations to determine the total revenue over the two days:
Monday: 3 T-shirts at $10 each, 4 hats at $15 each, and 1 pair of shorts at $20. Tuesday: 4 T-shirts at $10 each, 2 hats at $15 each, and 3 pairs of shorts at $20.
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