1 standards 2, 25 matrices and systems of equations introduction adding matrices multiplying...

1

Standards 2, 25 MATRICES AND SYSTEMS OF EQUATIONS

INTRODUCTION

ADDING MATRICES

MULTIPLYING MATRICES

INVERSE OF A MATRIX

IDENTITY MATRIX

SOLVING SYSTEMS WITH INVERSE MATRIX

SOLVING EQUATIONS WITH AUGMENTED MATRICES

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2

Standard 2:

Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

Estándar 2:

Los estudiantes resuelven sistemas de ecuaciones lineares y desigualdades (en 2 o tres variables) por substitución, con gráficas o con matrices.

Standard 25:

Students use properties from number systems to justify steps in combining and simplifying functions.

Estándar 25:

Los estudiantes usan propiedades de sistemas numéricos para justificar pasos en combinar y simplificar funciones.

ALGEBRA II STANDARDS THIS LESSON AIMS:

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3

Standards 2, 25

MATRICES

a 6

7 -2

5 y

x -2

columns

rowsB=

This matrix B has dimensions 2X4

326

C=

Matrix C is a column matrix of 3X1

5 -2 x zD=

Matrix D is a row matrix of 1X4

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4

Standards 2, 25

5 6

7 -22

10 12

14 -4=

2x+16y-4

52=

2x+1=5 6y-4=2

-1 -1

2x = 42 2

x= 2

+4 +4

6y = 66 6

y= 1

Solve the following problems involving matrices:

2 5

3 -4x

10 15

12 -4=

2x 5x

3x -4x

10 15

12 -4=

2x = 10 5x = 15 3x = 12 -4x = -4

2 2

x= 5

5 5

x= 3

3 3

x= 4

-4 -4

x= 1

Multiplying by one scalar:

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5

Standards 2, 25

e 5

4 -3E=

e 5

4 -3B=

Matrix E and matrix B have the same dimensions 2X2 and the same elements, so they are equal.

3 4

7 -9

f z

s -2F=

421

G=

Are matrices F and G equal? No, they have different number of columns and rows and different elements.

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6

Standards 2, 25

2 3

-2 5

4 6

7 10+ = 2+4 3+6

-2+7 5+10=

6 9

5 15

6 8

7 2

2 4

7 10- = 6-2 8-4

7-7 2-10=

4 4

0 -8

Observe that both matrices that are added or subtracted have the same dimensions.

3 -2 4 12 1 0 2 3+ = 6 -4 8 2 1 0 2 3+

7 -4 10 5 =

ADDING MATRICES

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7

Standards 2, 25

MULTIPLYING MATRICES

To multiply matrices the matrix at the left needs to have the same number of columns as rows have the one at the right, and the resulting matrix will have same number of rows as the one at the right and columns as the one at the left.

5 7 4

1 3 2

214

2X3 3X1

It is possible

2X1resulting matrix

(5)(2)+(7)(1)+(4)(4)(1)(2)+(3)(1)+(2)(4)

= =10 + 7 + 16

2 + 3 + 8=

33

13

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8

Standards 2, 25

MULTIPLYING MATRICES

2X3 3X2

It is possible

2X2resulting matrix

(3)(2)+(5)(1)+(1)(5)

(1)(2)+(3)(1)+(2)(5)=

6 + 5 + 5

2 + 3 + 10

3 5 1

1 3 2

215

426

(3)(4)+(5)(2)+(1)(6)

(1)(4)+(3)(2)+(2)(6)

=12 + 10 + 6

4 + 6 + 12

16 28

15 22=

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9

Standards 2, 25

8 2

6 4= (8)(4) –(6)(2) =32 -12=20

Calculate the A for matrix A:-1

8 2

6 4A=

A =-1 1 20

4 -2

-6 8=

420

-2 20

-6 20

820

=

1 5

-1 10

-3 10

2 5

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10

Standards 2, 25

4 3

5 1= (4)(1) –(5)(3) =4 -15 = -11

Calculate the A for matrix A:-1

4 3

5 1A=

A =-1 1-11

1 -3

-5 4=

-111

3 11

5 11

-411

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11

Standards 2, 25

IDENTITY MATRIX

A I = I A = A

1 0

0 1I=

8 2

6 4A=

1 0

0 1

8 2

6 4I A = = 1(8) + 0(6) 1(2) + 0(4)

0(8) + 1(6) 0(2) + 1(4)

8 2

6 4=

1 0

0 1

8 2

6 4A I = = 8(1) + 6(0) 2(1) + 4(0)

8(0) + 6(1) 2(0) + 4(1)

8 2

6 4=

Verify the identity property above indicated for matrix A below:

Diagonal

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12

Standards 2, 25

-3x + 6y = 157x + y = -8

Write the system of equations represented by each matrix equation:

-3 6

7 1

x

y=

15

-8

5x + 9y = 0-2x + 4y = 5

5 9

-2 4

x

y=

0

5

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13

Standards 2, 25Solve the following system of equations using matrices:

4x + 2y = 105x + y = 171

Write as matrix equation:

4 2

5 1

x

y=

10

17

4 2

5 1= (4)(1) –(5)(2) =4 -10 = -6

Finding the determinant of the

coefficient matrix:

Finding the inverse of the coefficient matrix:

1 -6

1 -2

-5 4=

-1 6

2 6

5 6

-4 6

=

-1 6

1 3

5 6

-2 3

4 2

5 1

x

y

10

17

-1 6

1 3

5 6

-2 3

=

-1 6

1 3

5 6

-2 3

Multiplying both sides by the inverse:

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14

Standards 2, 25Solve the following system of equations using matrices:

4x + 2y = 105x + y = 171

Write as matrix equation:

x

y

4 2

5 1=

10

17

x

y

4 2

5 1=

10

17

-1 6

1 3

5 6

-2 3

-1 6

1 3

5 6

-2 3

-1 6

1 3

(4) (5)+

5 6

-2 3

(4) (5)+

-1 6

1 3

(2) (1)+

5 6

-2 3

(2) (1)+

x

y

-1 6

1 3

(10) (17)+

5 6

-2 3

(10) (17)+=

1 0

0 1

x

y=

4

-3

Solution is (4,-3)

x

y=

4

-3

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15

Standards 2, 25Solve the following system of equations using matrices:

2x + 5y = 136x + 3y = 3

Write as matrix equation:

2 5

6 3

x

y=

13

3

2 5

6 3= (2)(3) –(6)(5) =6 -30 = -24

Finding the determinant of the

coefficient matrix:

Finding the inverse of the coefficient matrix:

1 -24

3 -5

-6 2=

-3 24

5 24

6 24

-2 24

=

-1 8

5 24

1 4

-112

2 5

6 3

x

y

13

3

-1 8

524

1 4

-112

=

-1 8

5 24

14

-1 12

Multiplying both sides by the inverse:

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16

Standards 2, 25Solve the following system of equations using matrices:

2x + 5y = 136x + 3y = 3

Write as matrix equation:

x

y

2 5

6 3=

13

3

x

y

2 5

6 3=

13

3

-1 8

524

1 4

-112

-1 8

524

1 4

-1 12

-18

5 24

(2) (6)+

1 4

-1 12

(2) (6)+

-1 8

5 24

(5) (3)+

1 4

-1 12

(5) (3)+

x

y

-1 8

5 24

(13) ( 3)+

1 4

-1 12

(13) ( 3)+=

1 0

0 1

x

y=

-1

3

Solution is (-1,3)

x

y=

-1

3

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17

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1.. (2)

13x

1-2 3

13

23

3 -2 1 2

2 3 -4 -4

4 2 -2 2

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18

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

x -2

=

-2(1) +2 = 0

-2( ) + 3 =-2 3

13 3

-2( ) - 4 = 1 3

-14 3

-2( ) - 4 = -16 3

2 3

+

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19

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

x -2

=

-2(1) +2 = 0

-2( ) + 1 =-2 3

7 3

-2( ) -1 = 1 3

- 5 3

-2( ) +1 = - 1 3

2 3

+

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20

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

0 -14 13

-16 13

1

3 -2 1 2

2 3 -4 -4

4 2 -2 2

313x

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21

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0

1

-15 39

-639

3 -2 1 2

2 3 -4 -4

4 2 -2 2

23x

=

2 3

-14 13

1 3

+ =-15 39

2 3

-16 13

2 3

+ =- 639

+

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22

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0 0

0

1

-15 39

-639

11 13

99 39

3 -2 1 2

2 3 -4 -4

4 2 -2 2

-7 3x

=

- 7 3

-14 13

5 3

- = 11 13

- 7 3

-16 13

1 3

- = 99 39

+

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23

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0 0

0

1

-15 39

-639

11 13

99 39

0 0 1 3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

1311x

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24

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0 0

0

1

-15 39

-639

11 13

99 39

0

0 0

1 0

1

2

3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

1413x

=

14 13

1613

- =3 2

+

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25

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0 0

0

1

-15 39

-639

11 13

99 39

1

0

0 0

0

1

0

0

1

1

2

3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

1539x

=

15 39

639

- =3 1+

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26

Standards 2, 25

3x- 2y + z = 22x+3y -4z = -4

4x+ 2y -2z = 2

Write the augmented matrix for this system, then reduce it to solve it:

3 -2 1 2

2 3 -4 -4

2 1 -1 1.. (2)

13x

1-2 3

13

23

0 13 3

-14 3

-16 3

0 7 3

-5 3

-1 3

1

0 -14 13

-16 13

0 0

0

1

-15 39

-639

11 13

99 39

1

0

0 0

0

1

0

0

1

1

2

3

3 -2 1 2

2 3 -4 -4

4 2 -2 2

-7 3x

23x

313x

1311x

x -2

1413x

1539x

=

=

=

=

=-2 +2 = 0

-2( ) + 3 =-2 3

13 3

-2( ) - 4 = 1 3

-14 3

-2( ) - 4 = -16 3

2 3

-2 +2 = 0

-2( ) + 1 =-2 3

7 3

-2( ) -1 = 1 3

- 5 3

-2( ) +1 = - 1 3

2 3

2 3

-14 13

1 3

+ =-15 39

2 3

-16 13

2 3

+ =- 639

- 7 3

-14 13

5 3

- = 11 13

- 7 3

-16 13

1 3

- = 99 39

14 13

1613

- =3 2

15 39

639

- =3 1

The solution is (1,2,3)

+

+

+

++

+=

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