252311921131213

25=2z+3y+x19=2z+y+x31=2z+y+3x

41009310

-19-521

4=z9=3z+y

-19=5z–2y+x

8.1 Matrices & Systems of Equations

• Augmented Matrix

• Row Operations

• Row Echelon Form –> Gaussian Elimination

• Reduced Row Echelon Form -> Gauss-Jordan Elimination

Augmented Matrix Example

Augmented Matrix in Row Echelon Form (Different Example)

* Reduced Row Echelon Form has 1’s along the diagonal and 0’s everywhere else

To solve a system of equations1.Write system of equations as an augmented matrix2.Use row operations to convert to row echelon form3.Find variables using back-substitution

WE’LL DO EXAMPLES IN CLASS!

Matrix Row Operations1. Two rows of a matrix may be interchanged.

2. The elements in any row may be multiplied by a nonzero number. (3R2)

3. Multiply a row by a non-zero number and add to another row

(Example: 2R1 + R3 -> R3)

-64-3-25-321

21-12183

Perform row operations:

Interchange: R1 R2 3R1 2R2 + R3R3

-64-3-221-121835-321

-64-3-25-321

63-36549

4-2105-321

21-12183

8.2 Matrix Algebra

A matrix of order m x n has m rows and n columns

A [aij] (Matrix a with elements aij)

Order: 2 x 3a23 = -1/5

a12. = 2

131

012

504

312

Matrix Addition

=

635

300

Scalar Multiplication

3 3 =

3 6 0

12 15 3 / 5

**Note: A matrix where all elements are zero is known as the zero matrix : 0

Equality of MatricesTwo matrices A and B are equal (A = B) if they have the sameOrder (number of rows and columns are equal) and each (i,j) entryOf A is equal to the corresponding (i,j) entry of B.

X – 8 6Y + 7 5 =

10 V15 R

x – 8 = 10 x = 18Y + 7 = 15 y = 8V = 6 and R = 5

X + Y 2 3 x - y =

4 23 1

x + y = 4X – y = 1So, x = 5/2 and y = 3/2

Matrix Multiplication

Note: AB is often not equal to BATry A = 2 3 X B = 3 4 -1 0 5 1

8.3 The Matrix InverseFor the Real Number System: A x 1 = A and 1 x A = ASo, 1 is the multiplicative Identity

For Matrices: A x I = A and I x A = ASo, I is the multiplicative Identity Matrix. I is a square matrix (2x2, 3x3, etc.) with 1’s in the Diagonal and 0’s elsewhere

5 7 1 0 5 7 1 0 5 76 8 0 1 6 8 0 1 6 8X = = X

For the Real Number System (A) (1/A) = 1 and (1/A)(A) = 1 (Identity)So, 1/A is the multiplicative inverse of A and A is the multiplicative inverse of 1/A

For Matrices: AA-1 = I and A-1A = I (If AB = BA = I then A & B are inverses)

-1 3 5 3 1 0 5 3 -1 3 2 -5 2 1 0 1 2 1 2 -5= =

Finding a Matrix Inverse or Proving None

To prove a matrix has no inverse, suppose has an inverse

Then,

Since the two matricies are equal y ouMust have 0 = 1, but this is False,Thus, No Inverse.

Find the multiplicative inverse of 2 1 5 3 2 1 w x 1 0 53 y z 0 1

2w + y = 1 2x + z = 05w + 3y = 0 5x + 3z = 1

Solve the system of equationsX = -1 w = 3 So, A-1 = 3 -1Y = -5 z = 2 -5 2

=

Special rule for inverse of a 2 x 2 matrixLet A= a b If ad – bc =0 the no inverse c d Otherwise, A-1 =

Another Method for Finding an Inverse

1 1 0 1 0 0

0 3 1 0 1 0

2 3 3 0 0 1

A I

Start With:

Perform Row Operations untilThe left is the Identity Matrix

2 1 1

6 3 11 1 0 0

8 8 82 3 1

0 1 08 8 86 1 3

0 0 18 8 8

R R R

��������������

1

6 3 1

8 8 8 6 3 12 3 1 1

2 3 18 8 8 8

6 1 36 1 3

8 8 8

A

Thus, the inverse is:

Solving Systems of Equations using InverseTo solve the system, solveThe matrix equation : AX = B X = A-1B

4

3 7

2 3 3 21

x y

y z

x y z

A X = B

Write the linear system in matrix form, then find the inverse. We know the inverse of this matrix since we found it in the previous example.

1

6 3 11

2 3 18

6 1 3

7

4

21

X A B

21

21

6 3 11

2 3 18

6 1

7

7

7

4

4

3 214

So, the solution set is {(3, 1, 4)}

8.4 Determinants and Cramer’s Rule

A a b

c d

The determinant of a 2 x 2 matrix, A is denoted det(A), |A| or

|A| = ad - bc

a b

c d

Example:

The determinant of a square n x n matrix, A, (n ≥ 3).is the sum of the entries in any row of A (or column of A), multiplied by their respective cofactors.

The minor Mij of the element aij is the determinant of the (n–1) × (n–1) matrix obtained by deleting the ith row and the jth column of A.

The cofactor Aij of the entry aij is given by: 1i j

ij ijA M

11

Finding Minors and Cofactors

a. Find minors M11 and M32

b. Find cofactors A11 and A32

c. Find |A|

To find M11 delete the first row and first columnThen, find the determinant: (-3)(7) – (-6)(1) = -21 - -6 = -21 + 6 = -15

To find M32 delete the third row and second columnThen, find the determinant: (-6)(2) – (4)(5) = -12 - 20 = -32

To find cofactors A11 & A32

A11 = (-1)1+1 (-15) = (-1)2 (-15) = 1(15) = 15A32 = (-1)3+2 (-32) = (-1)5 (-32) = -1(-32) = -32

1i j

ij ijA M

12

Example: Find the Determinant

Recall: The determinant of a square n x n matrix, is the sum of the entries in any row (or column), multiplied by their respective cofactors.

1i j

ij ijA M

13

© 2010 Pearson Education, Inc. All rights reserved

CRAMER’S RULE FOR SOLVINGTWO EQUATIONS IN TWO VARIABLES

a1x b1y c1

a2 x b2 y c2

The system

D a1 b1

a2 b2

,

x Dx

D and y

Dy

D

provided that D ≠ 0, where

Dx c1 b1

c2 b2

, and Dy a1 c1

a2 c2

.

two variables has a unique solution (x, y) given by

of two equations in

14© 2010 Pearson Education, Inc. All rights reserved

Using Cramer’s Rule to Solve Systems of Equations

Solve the system (2 x 2):

5 1

6

4

2 3

x y

x y

D 5 4

2 315 8 7

43 24 21.

3

1

6xD

530 2 28.

1

62yD

x Dx

D

21

7 3 and y

Dy

D

28

74.

1

4

7

0

5 2 3

4 3

x y z

x y z

x y z

Solve the system(3 x 3)

7 1 1

5 2 3

4 3 1

2 3 1 1 1 17 5 4

3 1 3 1 2 3

7 2 9 5 1 3 4 3 2

7 7 5 4 4 5 9

D

1i j

ij ijA M Recall:

1 1

2 3

3 1

2 3 1 1 1 1

3 1 3 1

1

2 3

1 2 9 4 1 3

1 7 4 4

0

1 4 0

9

4xD

1

4

0

1 4 0

7 1

5 3

4 1

5 3 7 1 7 1

4 1 4 1 5 3

1 5 12 4 7 4

17 44 27

yD

1

4

0

1 4 0

7 1

5 2

4 3

5 2 7 1 7 1

4 3 4 3 5 2

15 8 4 21 4

23 4 17 45

zD

x Dx

D

9

9 1

y Dy

D

27

93

z Dz

D

45

95

|A| = a11A11+a21A21+a31A31