Lesson7.2

Matrix Algebra

Precalculus

The points and are reflected across the given line.

Find the coordinates of the reflected points.

1. The -axis

2. The line

3. The line

Expand the ex

(

p

(a) (1, 3

ression,

4. sin( )

5. co

b )) ( ,

s

) x

x

y

x

y x

y

x y

( )x y

Quick Review

( ) (1,3)a ( ) ( , )b x y

( ) ( 3,1)a ( ) ( , )b y x

( ) (3, 1)a ( ) ( , )b y x

sin cos sin cosx y y x

cos cos sin sinx y x y

What you’ll learn about

MatricesMatrix Addition and SubtractionMatrix MultiplicationIdentity and Inverse MatricesDeterminant of a Square MatrixApplications

… 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 and be positive integers.

An (read " by matrix") is a rectangular

array of rows and columns of real numbers.

m n

m n

m n

matrixm×n

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a a

a a a

We also use the shorthand notation for this matrix.ija

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.

Matrix Vocabulary

Example Determining the Order of a Matrix

What is the order of the following matrix?

1 4 5

3 5 6

The matrix has 2 rows and 3 columns so

ordeit h r as 2 .3

Matrix Addition and Matrix Subtraction

Let and be matrices of order .ij ij

A a B b m n

1. The is the matrix

.ij ij

m n

A B a b

sum +A B

2. The is the matrix

.ij ij

m n

A B a b

difference A - B

Example Matrix Addition

1 2 3 2 3 4

4 5 6 5 6 7

2 3 4

5

1 2 3

4 65 76

2 3 4

5

1 2 3

4 5 6 6 7

3 5 7

9 11 13

Example Using Scalar Multiplication

1 2 33

4 5 6

3 6 9

12 15 18

31 2 3

4

3 3 3

3 53 63

The Zero Matrix

The matrix 0 [0] consisting

entirely of zeros is the .

m n zero matrix

0 0 0

0 0 0

Example:

Additive Inverse

Let be any matrix.

The matrix consisting of

the additive inverses of the entries of

is the because

0.

ij

ij

A a m n

m n B a

A

A B

additive inverse of A

Example Using Additive Inverse

Given matrix define below

2

0.5

6

find its additive inverse matrix ?

A

A

B

2 0

0.5 0

6 0

?

?

?

2

0.5

6

B

Matrix Multiplication

1 1 2 2

Let be any matrix and

be any matrix.

The product is the matrix

where + ... .i j i j

ij

ij

ij

ij ir rj

A a m r

B b r n

AB c m n

c a b a b a b

Example Matrix Multiplication

? ?

? ?AB

3 is 2A 2 is 3B

is 2 2 AB

1 1 2 3

?

0

?

2 ?

1 0 2 1 3

?

15

?

0 1 2

5 1

1 0 ?1

5 1

2 0 0 1 1 1 1

5 1

2 2

Find the product if possible.

1 01 2 3

and 2 1 0 1 1

0 1

AB

A B

Identity Matrix

The matrix with 1's on the main diagonal

and 0's elsewhere is the

.

1 0 0 0

0 1 0 0

0 0 1 0

0

0 0 0 0 1

n

n

n n I

I

identity matrix

of order

n n

Inverse of a Square Matrix

-1

Let be an matrix.

If there is a matrix such that

,

then is the of .

We write .

ij

n

A a n n

B

AB BA I

B A

B A

inverse

Example Inverse of a Square Matrices

Determine whether the matrices are inverses.

5 3 2 3,

3 2 3 5A B

1 0

0 1

10 9 15 15

6 6 9 10 AB

10 9 6 6

15 15

9 10BA

1 0

0 1

YES

Inverse of a 2 × 2 Matrix

1

If 0,

1then .

ad bc

a b d b

c d c aad bc

The number is the determinant

of the 2 2 matrix .

ad bc

a bA

c d

Determinant of a Square Matrix

Let be a matrix of order ( 2).

The determinant of , denoted by det or | | ,

is the sum of the entries in any row or any column

multiplied by their respective cofactors.

For example, expa

ijA a n n n

A A A

1 1 2 2

nding by the th row gives

det | | ... .i i i i in in

i

A A a A a A a A

Refer to text pg 583

Inverses of n × n Matrices

An n × n matrix A has an inverse if and only if det A ≠ 0.

Example Finding Inverse Matrices

1 3Find the inverse matrix if possible.

2 5A

1 1Use the formula

d bA

c aad bc

Since det 1 5 2 3 1 0,

must have an inverse.

A ad bc

A

5 3

2 1

15 3

1

2 11A

Properties of MatricesLet A, B, and C be matrices whose orders are such thatthe following sums, differences, and products are defined.

1. Commutative propertyAddition: A + B = B + AMultiplication: Does not hold in general

2. 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 thatthe following sums, differences, and products are defined.

4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠0

5. 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

Homework:

Text pg588/589 Exercises #2, 4, 14, 20, 24, and 34