8.1-8.58.1-8.58.1-8.58.1-8.5

Matrix AlgebraMatrix Algebra

Quick Review

The points (a) (1, 3) and (b) ( , ) are reflected across the given line.

Find the coordinates of the reflected points.

1. The -axis

2. The line

3. The line

Expand the expression,

4. sin( )

5. cos

x y

x

y x

y x

x y

( )x y

Quick Review Solutions The points (a) (1, 3) and (b) ( , ) are reflected across the given line.

Find the coordinates of the r

(a) (1,3) (b) ( ,

eflected points.

1. The -axis

2. The line

3.

)

(a) ( 3,

Th

1) (b) ( , )

e

x yx

x x

x

y y

y

line

Expand the expression,

4.

(a) ( 3, 1) (b) ( , )

sin cos sin cossin( )

5. cos( cos cos sin s n) i

y x

x y y x

x y x

y

y

x

x y

x y

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

11 12 1

21 22 2

1 2

Let and be positive integers. An (read " by matrix")

is a rectangular array of rows and columns of real numbers.

We also use the shorth

n

n

m m mn

m n m n

m n

a a a

a a a

a a a

matrix

m×n

and notation for this matrix.ija

Matrix VocabularyEach 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

What is the order of the following matrix?

1 4 5

3 5 6

The matrix has 2 rows and 3 columns so it has order 2 3.

Matrix Addition and Matrix Subtraction

Let and be matrices of order .

1. The is the matrix .

2. The is the matrix .

ij ij

ij ij

ij ij

A a B b m n

m n A B a b

m n A B a b

sum +

difference

A B

A - B

Example Matrix Addition

1 2 3 2 3 4

4 5 6 5 6 7

Example Matrix Addition

1 2 3 2 3 4

4 5 6 5 6 7

3 5 7

9 11 13

Example Using Scalar Multiplication

1 2 3

34 5 6

Example Using Scalar Multiplication

1 2 3

34 5 6

3 6 9

12 15 18

The Zero Matrix

The matrix 0 [0] consisting entirely of zeros is the .m n zero matrix

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

Matrix Multiplication

1 1 2 2

Let be any matrix and be any matrix.

The product is the matrix where

+ ... .

ij ij

ij

ij i j i j ir rj

A a m r B b r n

AB c m n

c a b a b a b

Example Matrix Multiplication

Find the product if possible.

1 01 2 3

and 2 1 0 1 1

0 1

AB

A B

Example Matrix Multiplication

11

12

The number of columns of is 3 and the number of rows of is 3,

so the product is defined. The product is a 2 2 matrix where

1

1 2 3 2 1 1 2 2 3 0 5,

0

0

1 2 3 1 1 0 2

1

ij

A B

AB c

c

c

21

22

1 3 1 1,

1

0 1 1 2 0 1 1 2 1 0 2,

0

0

0 1 1 1 0 0 1 1 1 1 2.

1

5 1Thus .

2 2

c

c

AB

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

Inverse of a 2 × 2 Matrix

1

1If 0, then .

a b d bad bc

c d c aad bc

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 ith row gives

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

A A a A a A a A

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 3

Find the inverse matrix if possible. 2 5

A

Example Finding Inverse Matrices

1 3

Find the inverse matrix if possible. 2 5

A

1

Since det 1 5 2 3 1 0, must have an inverse.

5 31 1Use the formula

2 11

5 3 .

2 1

A ad bc A

d bA

c aad bc

Properties of MatricesLet 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 = A4. 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

8.1-8.5 (cont.)8.1-8.5 (cont.)8.1-8.5 (cont.)8.1-8.5 (cont.)

Multivariate Linear Systems and Multivariate Linear Systems and Row OperationsRow Operations

Quick Review

3

1. Find the amount of pure acid in 45L of a 58%

acid solution.

2. Find the amount of water in 30 L of a 28%

acid solution.

3. Is the point (0, 1) on the graph of the function

( ) 4 1?

4. Solve for

f x x x

x

in terms of the other variables:

2

2 15. Find the inverse of the matrix .

0 3

x z w

Quick Review Solutions

3

1. Find the amount of pure acid in 45L of a 58%

acid solution.

2. Find the amount of water in 30 L of a 28%

acid solution.

3. Is the point (0, 1) on the graph of the function

26.1 L

21.6 L

( ) 4 1f x x x

?

4. Solve for in terms of the other variables:

2

2 15. Find the inverse of th

yes

2

1e matrix .

/2 1/ 6

0 1 30 3 /

x z

x

x wz w

What you’ll learn about• Triangular Forms for Linear Systems• Gaussian Elimination• Elementary Row Operations and Row Echelon Form• Reduced Row Echelon Form• Solving Systems with Inverse Matrices• Applications

… and whyMany applications in business and science are modeled by systems of linear equations in three or more variables.

Equivalent Systems of Linear Equations

The following operations produce an equivalent system of linear equations.1. Interchange any two equations of the

system.2. Multiply (or divide) one of the equations by

any nonzero real number.3. Add a multiple of one equation to any other

equation in the system.

Row Echelon Form of a Matrix

A matrix is in row echelon form if the following conditions are satisfied.1. Rows consisting entirely of 0’s (if there are

any) occur at the bottom of the matrix.2. The first entry in any row with nonzero entries

is 1.3. The column subscript of the leading 1 entries

increases as the row subscript increases.

Elementary Row Operations on a Matrix

A combination of the following operations will

transform a matrix to row echelon form.1. Interchange any two rows.2. Multiply all elements of a row by a

nonzero real number.3. Add a multiple of one row to any other

row.

Example Finding a Row Echelon Form

Solve the system:

2 3 1

5 3 10

3 6 5

x y z

x y z

x y z

Example Finding a Row Echelon Form

21 1 2

Apply elementary row operations to find a row echelon form of the augmented matrix.

2 3 1 1 1 5 3 10 1 5 3 10

1 5 3 10 2 3 1 1 2 0 13 5 21 3

3 1 6 5 3 1 6 5 3 1 6 5

R R R R

BBBBBBBBBBBBBBBBBBBBBBBBBBBB 1 3

2 2 3 3

1 5 3 10 1 5 3 101 5 3 10

1 5 21 5 21 130 13 5 21 0 1 14 0 1

13 13 13 13 13 310 14 3 25

0 14 3 25 31 310 0

13 13

1 5 3 10

5 210 1

13 130 0 1 1

R

R R R R

BBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

Convert the matrix to equations and solve by substitution.

1; 5 /13 21/13 so 2; 10 3 10 so 3.

The solution is 3, 2,1 .

z y y x x

Reduced Row Echelon Form

If we continue to apply elementary row operations to a row echelon form of a

matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the

reduced echelon form.

Example Solving a System Using

Inverse Matrices

Solve the system

2 3 0

2 2 10

x y

x y

Example Solving a System Using

Inverse Matrices

Solve the system

2 3 0

2 2 10

x y

x y

-1

Write the system as a matrix equation.

2 3 0Let , , and .

2 2 10

2 3 2 3Then so that

2 2 2 2

, where is the coefficient matrix of the system.

xA X B

y

x x yA X

y x y

AX B A

A

-1

exists since det 0. Use grapher to find

15. The solution of the system is (15,10).

10

A

X A B