copyright © 2011 pearson, inc. 7.2 matrix algebra

Copyright © 2011 Pearson, Inc.

7.2Matrix Algebra

2 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications

… and whyMatrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.


Matrix

Let m and n be positive integers. An m ×n matrix

(read "m by n matrix") is a rectangular array of

m rows and n columns of real numbers.

a11 a12 L a1n

a21 a22 L a2n

M M M

am1 am2 L amn

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

We also use the shorthand notation aij⎡⎣

⎤⎦ for this matrix.


Matrix Vocabulary

Each element, or entry, aij, of the matrix uses

double subscript notation. The row subscript is

the first subscript i, and the column subscript is

j. The element aij is the ith row and the jth

column. In general, the order of an m n

matrix is m n.


Example Determining the Order of a Matrix

What is the order of the following matrix?

1 4 5

3 5 6

⎡

⎣⎢

⎤

⎦⎥


Example Determining the Order of a Matrix

The matrix has 2 rows and 3 columns

so it has order 2×3.

What is the order of the following matrix?

1 4 5

3 5 6

⎡

⎣⎢

⎤

⎦⎥


Matrix Addition and Matrix Subtraction

Let A = aij⎡⎣

⎤⎦ and B= bij

⎡⎣

⎤⎦ be matrices of order m×n.

1. The sum A+ B is the m×n matrix

A+ B= aij +bij⎡⎣

⎤⎦.

2. The difference A−B is the m×n matrix

A−B= aij −bij⎡⎣

⎤⎦.


Example Matrix Addition

1 2 3

4 5 6

⎡

⎣⎢

⎤

⎦⎥+

2 3 45 6 7

⎡

⎣⎢

⎤

⎦⎥


Example Matrix Addition

A+ B=

2+1 2 +3 3+ 44 +5 5+6 6 +7

⎡

⎣⎢

⎤

⎦⎥=

3 5 79 11 13

⎡

⎣⎢

⎤

⎦⎥

1 2 3

4 5 6

⎡

⎣⎢

⎤

⎦⎥+

2 3 45 6 7

⎡

⎣⎢

⎤

⎦⎥


Example Using Scalar Multiplication

3

1 2 3

4 5 6

⎡

⎣⎢

⎤

⎦⎥


Example Using Scalar Multiplication

=

3⋅1 3⋅2 3⋅33⋅4 3⋅5 3⋅6

⎡

⎣⎢

⎤

⎦⎥=

3 6 912 15 18

⎡

⎣⎢

⎤

⎦⎥

3

1 2 3

4 5 6

⎡

⎣⎢

⎤

⎦⎥


The Zero Matrix

The m×n matrix 0 =[0] consisting entirely of

zeros is the zero matrix.


Additive Inverse

Let A = aij⎡⎣

⎤⎦ be any m×n matrix.

The m×n matrix B= aij⎡⎣

⎤⎦ consisting of the additive

inverses of the entries of A is the additive inverse of Abecause A+ B=0.


Matrix Multiplication

Let A = aij⎡⎣

⎤⎦ be any m×r matrix and B= bij

⎡⎣

⎤⎦

be any r ×n matrix.

The product AB= cij⎡⎣

⎤⎦ is the m×n matrix where

cij =ai1b1 j+ai2b2 j + ...+ airbrj .


Example Matrix Multiplication

Find the product AB if possible.

A =1 2 30 1 −1

⎡

⎣⎢

⎤

⎦⎥ and B=

1 02 10 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥



A =1 2 30 1 −1

⎡

⎣⎢

⎤

⎦⎥ and B=

1 02 10 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The number of columns of A is 3 and the number of

rows of B is 3, so the product is defined.

The product AB = cij⎡⎣

⎤⎦ is a 2×2 matrix where

c11 = 1 2 3⎡⎣ ⎤⎦

120

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=1⋅1+ 2⋅2 +3⋅0 =5,



A =1 2 30 1 −1

⎡

⎣⎢

⎤

⎦⎥ and B=

1 02 10 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

c12= 1 2 3⎡⎣ ⎤⎦

01−1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=1⋅0+ 2⋅1+3⋅−1=−1,

c21 = 0 1 −1⎡⎣ ⎤⎦

120

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=0⋅1+1⋅2 +−1⋅0 =2,



A =1 2 30 1 −1

⎡

⎣⎢

⎤

⎦⎥ and B=

1 02 10 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

c22= 0 1 −1⎡⎣ ⎤⎦

01−1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=0⋅0+1⋅1+−1⋅−1=2.

Thus AB=5 −12 2

⎡

⎣⎢

⎤

⎦⎥.


Identity Matrix

The n×n matrix In with 1's on the main diagonal and

0's elsewhere is the identity matrix of order n×n.

In =

1 0 0 L 00 1 0 L 00 0 1 L 0M M M 00 0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥


Inverse of a Square Matrix

Let A = aij⎡⎣

⎤⎦ be an n×n matrix.

If there is a matrix B such that

AB=BA=In,

then B is the inverse of A. We write B=A−1.


Inverse of a 2 × 2 Matrix

If ad −bc≠0, then

a bc d

⎡

⎣⎢

⎤

⎦⎥

−1

=1

ad−bcd −b−c a

⎡

⎣⎢

⎤

⎦⎥.


Determinant of a Square Matrix

Let A = aij⎡⎣

⎤⎦ be a matrix of order n×n (n> 2).

The determinant of A, denoted by detA or |A|,is the sum of the entries in any row or any column

multiplied by their respective cofactors. For

example, expanding by the ith row gives

detA=|A|=ai1Ai1 + ai2Ai2 + ...+ ainAin.


Inverses of n n Matrices

An n n matrix A has an inverse if and only if

det A ≠ 0.


Example Finding Inverse Matrices

Determine whether the matrix has an inverse.

If so, find its inverse matrix.

A =5 18 3

⎡

⎣⎢

⎤

⎦⎥



Since det A =ad−bc=5⋅3−1⋅8 =7 ≠0,we conclude that A has an inverse.

Use the formula A−1 =1

ad−bcd −b−c a

⎡

⎣⎢

⎤

⎦⎥=

17

3 −1−8 5

⎡

⎣⎢

⎤

⎦⎥

=

37

−17

−87

57

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

.

A =

5 18 3

⎡

⎣⎢

⎤

⎦⎥



Check:

A−1A=

37

−17

−87

57

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

5 18 3

⎡

⎣⎢

⎤

⎦⎥=

37−87

37−37

−407

+407

−87+157

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=1 00 1

⎡

⎣⎢

⎤

⎦⎥=I2

A =

5 18 3

⎡

⎣⎢

⎤

⎦⎥

Similarly, A−1A=I2 .


Properties of Matrices

Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.1. Community propertyAddition: A + B = B + AMultiplication: Does not hold in general2. Associative propertyAddition: (A + B) + C = A + (B + C)Multiplication: (AB)C = A(BC)3. Identity propertyAddition: A + 0 = AMultiplication: A·In = In·A = A


Properties of Matrices

Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠05. Distributive propertyMultiplication over addition:A(B + C) = AB + AC (A + B)C = AC + BCMultiplication over subtraction:A(B – C) = AB – AC (A – B)C = AC – BC


Quick Review

The points (a) (1,−3) and (b) (x, y) are reflectedacross the given line.Find the coordinates of the reflected points.1. The x-axis

2. The line y=x3. The line y=−xExpand the expression,4. sin(x+ y)5. cos(x+ y)


Quick Review Solutions

The points (a) (1,−3) and (b) (x, y) are reflectedacross the given line.Find the coordinates of the reflected points.1. The x-axis (a) (1,3) (b) (x,−y)2. The line y=x (a) ( −3,1) (b) (y,x)3. The line y=−x (a) ( −3,−1) (b) (−y,−x)Expand the expression,4. sin(x+ y) sinxcosy+sinycosx5. cos(x+ y) cosxcosy−sinxsiny

copyright © 2011 pearson, inc. 7.2 matrix algebra

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