copyright 2013, 2010, 2007, pearson, education, inc. section 7.3 matrices

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 7.3

Matrices


What You Will Learn

Introduction to MatricesAddition of MatricesSubtraction of MatricesMultiplying a Matrix by a Real NumberMultiplication of Matrices

7.3-2


MatrixA matrix is a rectangular array of elements.

An array is a systematic arrangement of numbers or symbols in rows and columns.

7.3-3


MatrixMatrices (the plural of matrix) may be used to display information and to solve systems of linear equations.

The numbers in the rows and columns of a matrix are called the elements of the matrix.

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Dimensions of a MatrixThe dimensions of a matrix may be indicated with the notation r × s, where r is the number of rows and s is the number of columns of a matrix.This is a 2 × 4 matrix.

102 93 22 3582 94 23 49

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Dimensions of a MatrixA matrix that contains the same number of rows and columns is called a square matrix.

2 × 2square matrix

2 35 2

3 × 3square matrix

4 0 24 1 36 8 9

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Equal Matrices

Two matrices are equal if and only if they have the same elements in the same relative positions.

4 0 24 1 36 8 9

4 0 24 1 36 8 9

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Addition and Subtraction of MatricesTwo matrices can only be added or subtracted if they have the same dimensions.The corresponding elements of the two matrices are either added or subtracted.

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Example 4: Subtracting MatricesDetermine A – B if

A 3 6

5 1

and B 2 4

8 3

Solution

A B 3 6

5 1

2 4

8 3

3 2 6 ( 4)

5 8 1 ( 3)

1 10

3 2

7.3-9


Multiplying a Matrix by a Real NumberA matrix may be multiplied by a real number, a scalar, by multiplying each entry in the matrix by the real number.

A

e f g

h i j

3A 3e 3f 3g

3h 3i 3 j

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Multiplication of Matrices

Multiplication of matrices is possible only when the number of columns in the first matrix is the same as the number of rows of the second matrix.For example, we can multiply a 2 × 2 matrix times a 2 × 3 matrix.

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Multiplication of MatricesIn general,

A B a b

c d

e f g

h i j

ae bh af bi ag bj

ce dh cf di cg dj

Note that Matrix Multiplication is not commutative: A × B ≠ B × A.7.3-12


3(5) 1(2) 3( 1) 1(8) 3(3) 1(0)

4(5) 2(2) 4( 1) 2(8) 4(3) 2(0)

A B 3 1

4 2

5 1 32 8 0

Example 7: Multiplying MatricesDetermine A × B if

A 3 1

4 2

and B 5 1 3

2 8 0

Solution

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Multiplicative Identity MatrixSquare matrices have a multiplicative identity matrix.Note that in any multiplicative identity matrix, 1’s go diagonally from top left to bottom right and all other elements in the matrix are 0’s.

I 1 0

0 1

I 1 0 00 1 00 0 1

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Multiplicative Identity MatrixFor any square matrix A,

A × I = I × A = A.

If then,

A 4 3

2 1

A I 4 3

2 1

1 00 1

4 3

2 1

A

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Example 9: A Manufacturing ApplicationThe Fancy Frock Company manufactures three types of women’s outfits: a dress, a two-piece suit (skirt and jacket), and a three-piece suit (skirt, jacket, and a vest). On a particular day, the firm produces 20 dresses, 30 two-piece suits, and 50 three-piece suits. Each dress requires 4 units of material and 1 hour of work

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Example 9: A Manufacturing Applicationto produce, each two-piece suit requires 5 units of material and 2 hours of work to produce, and each three-piece suit requires 6 units of material and 3 hours to produce. Use matrix multiplication to determine the total number of units of material and the total number of hours needed for that day’s production.

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Example 9: A Manufacturing ApplicationSolution

A 20 30 50

DressTwo-piece

Three-piece

B 4 15 26 3

MaterialHours

Dress

Two-piece

Three-piece

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Example 9: A Manufacturing ApplicationSolutionThe product of A and B, will give the total number of units of material and the total number of hours of work needed for that day’s production.

A B 20 30 50

4 15 26 3

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Example 9: A Manufacturing ApplicationSolution

A B 20 30 50

4 15 26 3

20(4) 30(5) 50(6) 20(1) 30(2) 50(3)

530 230 A total of 530 units of material and a total of 230 hours of work are needed that day.7.3-20

copyright 2013, 2010, 2007, pearson, education, inc. section 7.3 matrices

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