p2a.1.3 matrices vocabulary - mrsfruge.com · matrix ! equals the number of rows in matrix $. ! ×...

P2A.1.3 Matrices Vocabulary

System

A set of two or more equations that form a solution point or area

Linear Function Each term has an exponent of one and the graphing of the equation results

in a straight line

Solution The value that when substituted for the variable in a given

equation/expression produces a true statement

Elimination (aka Gaussian and Back

Substitution)

A process used to solve systems of equations by combining two equations in a way that cancels a variable

Substitution A process used to solve a system of equations by replacing a variable in

one equation with an equivalent expression from the other equation

Matrix (Matrices)

A rectangular array of quantities organized by rows and columns

Rows

The horizontal in a matrix

Columns

The vertical in a matrix

Inverse of a Matrix The matrix must be square in order to have an inverse; inverse is denoted

as 𝐴"#

Ordered Triple

The solution of a linear equation of 3 variables

Order of a Matrix (aka Dimensions)

The number of rows by the number of columns

Entry

Each value in a matrix

Address The location of an entry in a matrix, expressed by using the lowercase

matrix letter with row and column numbers as subscripts

Scalar

A quantity that has a magnitude but no direction, such as a number

Matrix Multiplication

For matrices 𝐴 and 𝐵, 𝐴 × 𝐵 is defined only if the number of columns in matrix 𝐴 equals the number of rows in matrix 𝐵. 𝐴 × 𝐵 ≠ 𝐵 × 𝐴

Determinant

A scalar value that can be computed from the elements of a square matrix and determines certain properties of the linear transformation described by the matrix

Cramer’s Rule A formula for the solution of a system of linear equations with as many

equations as unknowns, valid whenever the system has a unique solution

Possible Solutions of a System

1. One unique solution 2. No solution 3. Infinite solutions

Identity Matrix

A square matrix that has 1’s on the main diagonal and 0’s everywhere else

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4-1

2nd Pass

Chapter 4 33 North Carolina StudyText, Math BC, Volume 2

Study GuideIntroduction to Matrices

Organize and Analyze Data

A matrix can be described by its dimensions. A matrix with m rows and n columns is an m × n matrix.

Owls’ eggs incubate for 30 days and their fledgling period is also 30 days. Swifts’ eggs incubate for 20 days and their fledgling period is 44 days. Pigeon eggs incubate for 15 days, and their fledgling period is 17 days. Eggs of the king penguin incubate for 53 days, and the fledgling time for a king penguin is 360 days. Write a 2 × 4 matrix to organize this information. Source: The Cambridge Factfinder

Owl Swift Pigeon King Penguin

Incubation ⎡ ⎢

⎣ 30 30 20

44 15

17 53

360

⎤

�

⎦

Fledgling

What are the dimensions of matrix A if A = ⎡ ⎢

⎣ 13

2 10

8 -3

15 45

80 ⎤ �

⎦ ?

Since matrix A has 2 rows and 4 columns, the dimensions of A are 2 × 4.

ExercisesState the dimensions of each matrix.

1.

⎡

⎢

⎣

15

23

14

63

5

6

70

3

27

0

24

42

-4

5

-3

90

⎤

�

⎦

2. [16 12 0] 3. ⎡ ⎢

⎢ ⎢ ⎢ ⎣ 71

39

45

92 78

44

27

16

53 65

⎤ �

� � � ⎦

4. A travel agent provides for potential travelers the normal high temperatures for the months of January, April, July, and October for various cities. In Boston these figures are 36°, 56°, 82°, and 63°. In Dallas they are 54°, 76°, 97°, and 79°. In Los Angeles they are 68°, 72°, 84°, and 79°. In Seattle they are 46°, 58°, 74°, and 60°. In St. Louis they are 38°, 67°, 89°, and 69°. Organize this information in a 4 × 5 matrix. Source: The New York Times Almanac

Matrixa rectangular array of variables or constants in horizontal rows and vertical columns, usually

enclosed in brackets.

Example 1

Example 2

Matrixa rectangular array of variables or constants in horizontal rows and vertical columns, usually

enclosed in brackets.

Example 1

Example 2

4-1 SCS MBC.N.2.1

0033_0056_ALG2_NC_S_C04_V2_8906933 330033_0056_ALG2_NC_S_C04_V2_8906933 33 3/16/10 6:07:00 PM3/16/10 6:07:00 PM

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2nd Pass


Study Guide (continued)

Introduction to Matrices

Elements of a Matrix A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns. The values are called elements and are identified by their location in the matrix. The location of an element is written as a subscript with the number of its row followed by the number of its column. For example, a12 is the element in the first row and second column of matrix A.

In the matrices below, 11 is the value of a12 in the first matrix. The value of b32 in the second matrix is 7.

A =

⎡

⎢

⎣

7

5

9 11

4

3

2

10

6

8

1

12 ⎤

�

⎦

Find the value of c23.

C = ⎡

⎢ ⎣ 2 3 5 4 3 1 ⎤

� ⎦

Since c23 is the element in row 2,column 3, the value of c23 is 1.

B =

⎡ ⎢

⎢

⎢

⎢ ⎣

3

5

8

11

4

9

10

7

13

2

12

15

6

1

14 ⎤ �

�

�

� ⎦

Find the value of d54.

matrix D =

⎡ ⎢

⎢

⎢

⎢ ⎣

25

7 17

22 5

11

8 6

16 23

4

9 15

21 3

1

12 18

24 14

20

13 2

19 10

⎤ �

�

�

� ⎦

Since d54 is the element in row 5, column 4, the value of d54 is 14.

Example 1 Example 2

F =

⎡

⎢

⎣

12

9

6

1

7

2 14

4

5

11 8

3

⎤

�

⎦

, G = ⎡ ⎢

⎢

⎢

⎢ ⎣

1 2

3

4

5

14

15 16

17 6

13

20 19

18 7

12

11 10

9 8

⎤ �

�

�

� ⎦ ,

H = ⎡

⎢

⎣

5

3

8

9 7

2

11

2 6

4

10 1 ⎤

�

⎦

.

1. f32 2. g51 3. h22

4. g43 5. h34 6. f23

7. h14 8. f42 9. g14

Exercises

Identify each element for the following matrices.

4-1 SCS MBC.N.2.1


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4-2

2nd Pass


4-2 Study GuideOperations with Matrices

Add and Subtract Matrices Matrices with the same dimensions can be added together or one can be subtracted from the other.

Find A + B if A = ⎡ ⎢

⎣ 6 2 -7

-12 ⎤ �

⎦ and B =

⎡ ⎢

⎣ 4 -5

2 -6

⎤ �

⎦ .

A + B = ⎡ ⎢ ⎣ 6 2 -7

-12 ⎤

� ⎦ +

⎡

⎢ ⎣ 4 -5

2 -6

⎤

� ⎦

= ⎡

⎢ ⎣ 6 + 4

2 + (-5)

-7 + 2

-12 + (-6)

⎤

� ⎦

= ⎡

⎢ ⎣ 10 -3

-5 -18

⎤

� ⎦

Find A - B if A = ⎡

⎢

⎣

-2

3 10

8

-4 7

⎤

�

⎦

and B = ⎡

⎢

⎣

4

-2 -6

-3

1 8 ⎤

�

⎦

.

A - B = ⎡

⎢

⎣

-2

3 10

8

-4

7 ⎤

�

⎦

- ⎡

⎢

⎣

4

-2

-6

-3

1

8 ⎤

�

⎦

= ⎡

⎢

⎣

-2 - 4

3 - (-2)

10 - (-6)

8 - (-3)

-4 - 1 7 - 8

⎤

�

⎦

= ⎡

⎢

⎣

-6

5 16

11

-5 -1

⎤

�

⎦

ExercisesPerform the indicated operations. If the matrix does not exist, write impossible.

1. ⎡

⎢ ⎣ 8 -10

7 -6

⎤

� ⎦ -

⎡

⎢ ⎣ -4

2 3

-12 ⎤ � ⎦ 2.

⎡ ⎢

⎣ 6 -3

-5 4 9 5 ⎤ � ⎦ +

⎡ ⎢ ⎣ -4

6 3 9 2

-4 ⎤

� ⎦

3. ⎡

⎢ ⎣

6

-3

2 ⎤

�

⎦

+ [-6 3 -2] 4. ⎡

⎢ ⎣

5

-4

7

-2

6

9 ⎤

�

⎦

+ ⎡

⎢

⎣

-11

2 -4

6

-5

-7 ⎤

�

⎦

5. ⎡

⎢ ⎣

8

4

-7

0 5

3

-6

-11

4 ⎤

�

⎦

- ⎡

⎢

⎣

-2

3 -8

1

-4

5 7 3

6 ⎤

�

⎦

6. ⎡

⎢ ⎣

3 − 4

- 1 − 2 2 − 5

4 − 3 ⎤

�

⎦

- ⎡

⎢

⎣

1 − 2

2 − 3

2 − 3

- 1 − 2 ⎤

�

⎦

Addition of Matrices

⎡

⎢

⎣

a

d

g b

e

h c f

i

⎤

�

⎦

+

⎡

⎢

⎣

j

m

p

k

n

q

l o

r

⎤

�

⎦

=

⎡

⎢

⎣

a + j

d + m g + p

b + k

e + n

h + q

c + l

f + o

i + r

⎤

�

⎦

Subtraction of Matrices

⎡

⎢

⎣

a

d

g b

e

h c f

i

⎤

�

⎦

-

⎡

⎢

⎣

j

m

p

k

n

q

l o

r

⎤

�

⎦

=

⎡

⎢

⎣

a - j

d - m g - p

b - k

e - n h - q

c - l

f - o

i - r

⎤

�

⎦

Example 1

Example 2

SCS MBC.N.2.2


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2nd Pass


4-2 Study Guide (continued)

Operations with Matrices

Scalar Multiplication You can multiply an m × n matrix by a scalar k.

If A = ⎡ ⎢ ⎣ 4 -6

0 3 ⎤ � ⎦ and B =

⎡ ⎢ ⎣ -1

7 5

8 ⎤ � ⎦ , find 3B - 2A.

3B - 2A = 3 ⎡

⎢ ⎣ -1

7 5

8 ⎤

� ⎦ - 2

⎡ ⎢ ⎣ 4 -6

0 3 ⎤

� ⎦ Substitution

= ⎡

⎢ ⎣ 3(-1)

3(7)

3(5)

3(8)

⎤

� ⎦ -

⎡

⎢ ⎣

2(4)

2(-6) 2(0)

2(3)

⎤

� ⎦ Multiply.

= ⎡ ⎢ ⎣ -3

21

15 24

⎤

� ⎦ -

⎡ ⎢ ⎣ 8 -12

0 6

⎤

� ⎦ Simplify.

= ⎡ ⎢ ⎣ -3 21

- 8 - (-12)

15 - 0 24 - 6

⎤

� ⎦ Subtract.

= ⎡ ⎢ ⎣ -11

33

15 18

⎤

� ⎦ Simplify.

ExercisesPerform the indicated operations. If the matrix does not exist, write impossible.

1. 6 ⎡

⎢

⎣

2

0

-4 -5

7 6

3

-1

9 ⎤

�

⎦

2. - 1 − 3 ⎡

⎢

⎣

6

51

-18

15

-33

3

9

24 45

⎤

�

⎦

3. 0.2 ⎡

⎢

⎣

25

5 60

-10

55 35

-45

-30 -95

⎤

�

⎦

4. 3 ⎡

⎢ ⎣ -4

2 5

3 ⎤

� ⎦ - 2

⎡

⎢ ⎣ -1

-3

2 5 ⎤ � ⎦ 5. -2

⎡

⎢ ⎣ 3 0 -1

7 ⎤ � ⎦ + 4

⎡

⎢ ⎣ -2

2 0 5 ⎤ � ⎦

6. 2 ⎡

⎢ ⎣ 6 -5

-10 8 ⎤ � ⎦ + 5

⎡

⎢ ⎣ 2 4 1

3 ⎤

� ⎦ 7. 4

⎡

⎢ ⎣ 1 -3

-2 4 5 1 ⎤ � ⎦ - 2

⎡

⎢ ⎣ 4 2 3

-5 -4

-1 ⎤

� ⎦

8. 8 ⎡

⎢

⎣

2

3

-2

1

-1

4 ⎤

�

⎦

+ 3 ⎡

⎢

⎣

4

-2

3

0

3

-4 ⎤

�

⎦

9. 1 − 4 (

⎡

⎢ ⎣ 9 -7

1 0

⎤

� ⎦ +

⎡ ⎢ ⎣ 3 1

-5 7 ⎤

� ⎦ )

Scalar Multiplication k ⎡

⎢

⎣ a

d

b

e c f ⎤

�

⎦ =

⎡

⎢

⎣ ka

kd

kb

ke

kc

kf

⎤

�

⎦

Example

SCS MBC.N.2.2


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4-3

2nd Pass


4-3 Study GuideMultiplying Matrices

Multiply Matrices You can multiply two matrices if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Find AB if A = ⎡

⎢

⎣

-4

2 1

3

-2 7 ⎤

�

⎦

and B = ⎡ ⎢

⎣ 5 -1

-2 3 ⎤ �

⎦ .

AB = ⎡

⎢

⎣

-4

2 1

3

-2

7 ⎤

�

⎦

· ⎡

⎢ ⎣ 5 -1

-2 3

⎤

� ⎦ Substitution

= ⎡

⎢

⎣

- 4(5) + 3(-1)

2(5) + (-2)(-1)

1(5) + 7(-1)

-4(-2) + 3(3)

2(-2) + (-2)(3)

1(-2) + 7(3)

⎤

�

⎦

Multiply columns by rows.

= ⎡

⎢

⎣

-23

12 -2

17

-10

19 ⎤

�

⎦

Simplify.

ExercisesFind each product, if possible.

1. ⎡

⎢ ⎣ 4 -2

1 3 ⎤

� ⎦ ·

⎡

⎢ ⎣ 3 0 0

3 ⎤ � ⎦ 2.

⎡

⎢ ⎣ -1

3 0

7 ⎤

� ⎦ ·

⎡

⎢ ⎣ 3 -1

2 4 ⎤

� ⎦ 3.

⎡

⎢ ⎣ 3 2 -1

4 ⎤

� ⎦ ·

⎡ ⎢ ⎣ 3 2

-1 4

⎤

� ⎦

4. ⎡

⎢ ⎣ -3

5 1

-2 ⎤

� ⎦ ·

⎡

⎢ ⎣ 4 -3

0 1 -2

1 ⎤

� ⎦ 5.

⎡

⎢ ⎣

3

0

-5

-2

4

1 ⎤

�

⎦

· ⎡

⎢ ⎣ 1 2 2

1 ⎤

� ⎦ 6.

⎡

⎢ ⎣ 5 2 -2

-3 ⎤

� ⎦ ·

⎡ ⎢ ⎣ 4 -2

-1 5

⎤

� ⎦

7. ⎡

⎢

⎣

6

-4

-2

10

3

7 ⎤

�

⎦

· [0 4 -3] 8. ⎡

⎢ ⎣ 7 5 -2

-4 ⎤

� ⎦ ·

⎡ ⎢ ⎣ 1 -2

-3 0 ⎤

� ⎦ 9.

⎡

⎢

⎣

2

1

-1

0 4

3

-3

-2

1 ⎤

�

⎦

· ⎡

⎢

⎣

2

3

-2 -2

1 4 ⎤

�

⎦

Multiplication of Matrices

A · B = AB

⎡

⎢

⎣ a

c

b

d ⎤

�

⎦ ·

⎡

⎢

⎣ e

g

f

h ⎤ �

⎦ =

⎡

⎢

⎣ ae + bg

ce + dg

af + bh

cf + dh

⎤

�

⎦

Example

SCS MBC.N.2.3


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2nd Pass



Multiplying Matrices

Multiplicative Properties The Commutative Property of Multiplication does not hold for matrices.

Use A = ⎡ ⎢

⎣ 4 2

-3 1 ⎤ �

⎦ , B =

⎡ ⎢

⎣ 2 5

0 -3

⎤ �

⎦ , and C =

⎡ ⎢

⎣ 1 6 -2

3 ⎤ �

⎦ to find each product.

a. (A + B)C

(A + B) C = ( ⎡ ⎢ ⎣ 4 2 -3

1 ⎤ �

⎦ +

⎡

⎢ ⎣ 2 5 0

-3 ⎤ �

⎦ ) ·

⎡

⎢ ⎣ 1 6

-2 3 ⎤ �

⎦

= ⎡

⎢ ⎣ 6 7

-3 -2

⎤ �

⎦ ·

⎡

⎢ ⎣ 1 6 -2

3 ⎤ �

⎦

= ⎡

⎢ ⎣ 6(1) + (-3)(6)

7(1) + (-2)(6)

6(-2) + (-3)(3)

7(-2) + (-2)(3)

⎤ �

⎦

= ⎡

⎢ ⎣ -12

-5 -21

-20 ⎤ �

⎦

b. AC + BC AC + BC =

⎡

⎢ ⎣ 4 2 -3

1 ⎤

� ⎦ ·

⎡

⎢ ⎣ 1 6 -2

3 ⎤

� ⎦ +

⎡

⎢ ⎣ 2 5 0

-3 ⎤

� ⎦ ·

⎡

⎢ ⎣ 1 6 -2

3 ⎤

� ⎦

= ⎡

⎢ ⎣ 4(1) + (-3)(6)

2(1) + 1(6)

4(-2) + (-3)(3)

2(-2) + 1(3)

⎤

� ⎦ +

⎡

⎢ ⎣ 2(1) + 0(6)

5(1) + (-3)(6)

2(-2) + 0(3)

5(-2) + (-3)(3)

⎤

� ⎦

= ⎡

⎢ ⎣ -14

8 -17

-1 ⎤

� ⎦ +

⎡

⎢ ⎣ 2 -13

-4 -19

⎤

� ⎦ =

⎡

⎢ ⎣ -12

-5 -21

-20 ⎤

� ⎦

Note that although the results in the example illustrate the Right Distributive Property, they do not prove it.

Exercises

Use A = ⎡ ⎢

⎣ 3 5

2 -2

⎤ �

⎦ , B =

⎡ ⎢

⎣ 6 2 4

1 ⎤ �

⎦ , C =

⎡

⎢

⎣

- 1 − 2

1

-2

-3 ⎤

�

⎦

, and scalar c = -4 to determine whether

the following equations are true for the given matrices.

1. c(AB) = (cA)B 2. AB = BA

3. BC = CB 4. (AB)C = A(BC)

5. C(A + B) = AC + BC 6. c(A + B) = cA + cB

Properties of Matrix MultiplicationFor any matrices A, B, and C for which the matrix product is

defined, and any scalar c, the following properties are true.

Associative Property of Matrix Multiplication (AB)C = A(BC)

Associative Property of Scalar Multiplication c(AB) = (cA)B = A(cB)

Left Distributive Property C(A + B) = CA + CB

Right Distributive Property (A + B)C = AC + BC

Example

SCS MBC.N.2.3


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Less

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4-5

2nd Pass


Determinants A 2×2 matrix has a second-order determinant; a 3×3 matrix has a third-order determinant.

Evaluate each determinant.

a. ⎪ 6 -8

3 5

⎥

⎪ 6 -8

3 5

⎥ = 6 (5) - 3 (-8)

= 54

ExercisesEvaluate each determinant.

1. ⎪ 6 5 -2

7 ⎥ 2. ⎪ 3

9 2

6 ⎥ 3. ⎪

3

0

-1

-2

4

4

-2

1 -3

⎥ 4. Find the area of a triangle with vertices (2, –3), (7, 4), and (–5, 5).

Second-Order

DeterminantFor the matrix

⎡ ⎢ ⎣ a

c

b

d ⎤

� ⎦ , the determinant is ⎪ a

c

b

d ⎥ = ad – bc.

Third-Order

Determinant

For the matrix

⎡

⎢

⎣

a d

g

b e

h

c f

i

⎤

�

⎦

, the determinant is found using the diagonal rule.

⎡

⎢

⎣

a

d

g

b e

h

c f

i

⎤

�

⎦

a

d

g

b

e h

⎡

⎢

⎣

a d

g

b e

h

c f

i

⎤

�

⎦

a

d

g

b

e h

Area of a Triangle

The area of a triangle having vertices (a, b), (c, d ), and (e, f ) is ⎪A⎥ ,

where A = 1 − 2 ⎪

a

c

e

b d

f

1

1

1

⎥ .

Example

4-5 Study GuideDeterminants and Cramer’s Rule

b. ⎪ 4

1

2

5

3

-3

-2

0 6

⎥

⎪ 4 1

2

5

3

-3

-2

0

6 ⎥

4 1

2

5

3

-3 ⎪

4 1

2

5

3

-3

-2

0

6 ⎥

4

1 2

5

3

-3

= [4(3)6 + 5(0)2 + (-2)1(-3)] - [(-2)3(2) + 4(0)(-3) + 5(1)6]

= [72 + 0 + 6] - [-12 + 0 + 30]

= 78 - 16 or 60

SCS MBC.N.2.1, MBC.A.2.1


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2nd Pass


Cramer’s Rule Determinants provide a way for solving systems of equations.

Use Cramer’s Rule to solve the system of equations. 5x - 10y = 810x + 25y = -2

x = ⎪ m n

b g ⎥ −

⎪C⎥ Cramer’s Rule y =

⎪ a f m n

⎥ −

⎪C⎥

=

⎪ 8

-2 -10

25 ⎥ −

⎪ 5 10

-10 25

⎥ a = 5, b = -10, f = 10, g = 25, m = 8, n = -2 =

⎪ 5 10

8 -2

⎥ −

⎪ 5 10

-10 25

⎥

= 8(25) - (-2)(-10) −−

5(25) - (-10)(10) Evaluate each determinant. = 5(-2) - 8(10)

−− 5(25) - (-10)(10)

= 180 − 225

or 4 − 5 Simplify. = - 90 −

225 or - 2 −

5

The solution is ( 4 − 5 , - 2 −

5 ) .

ExercisesUse Cramer’s Rule to solve each system of equations. 1. 3x - 2y = 7 2. x - 4y = 17 3. 2x - y = -2

2x + 7y = 38 3x - y = 29 4x - y = 4

4. 2x - y = 1 5. 4x + 2y = 1 6. 6x - 3y = -35x + 2y = -29 5x - 4y = 24 2x + y = 21

7. 2x + 7y = 16 8. 2x - 3y = -2 9. x − 3 +

y −

5 = 2

x - 2y = 30 3x - 4y = 9 x −

4 -

y −

6 = -8

10. 6x - 9y = -1 11. 3x - 12y = -14 12. 8x + 2y = 3 − 7

3x + 18y = 12 9x + 6y = -7 5x - 4y = - 27 − 7

Cramer’s Rule for

Two-Variable Systems

Let C be the coefficient matrix of the system ax + by = m →

fx + gy = n

The solution of this system is x =

⎪ m n

b

g

⎥ −

⎪ C ⎥ , y =

⎪ a f

m

n ⎥ −

⎪C ⎥ , if C ≠ 0.

Example

Study Guide (continued)

Determinants and Cramer’s Rule

4-5

⎪ a f

b

g ⎥



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Less

on

4-6

1st Pass


Identity and Inverse Matrices The identity matrix for matrix multiplication is a square matrix with 1s for every element of the main diagonal and zeros elsewhere.

If an n × n matrix A has an inverse A-1, then A � A-1 = A-1 � A = I.

Determine whether X = ⎡ ⎢

⎣ 7 10

4 6 ⎤ �

⎦ and Y =

⎡

⎢

⎣

3 -5

-2 7 − 2

⎤

�

⎦

are inverse matrices.

Find X · Y.

X · Y = ⎡ ⎢

⎣ 7 10

4 6 ⎤ �

⎦ ·

⎡ ⎢

⎢

⎣ 3 -5

-2 7 − 2 ⎤ �

�

⎦

= ⎡ ⎢

⎣ 21 - 20 30 - 30

-14 + 14

-20 + 21

⎤ �

⎦ or

⎡ ⎢

⎣ 1 0

0 1 ⎤ �

⎦

Find Y · X.

Y · X = ⎡ ⎢

⎢

⎣ 3 -5

-2 7 − 2 ⎤ �

�

⎦ ·

⎡ ⎢

⎣ 7 10

4 6 ⎤ �

⎦

= ⎡ ⎢

⎣ 21 - 20

-35 + 35 12 - 12

-20 + 21 ⎤ �

⎦ or

⎡ ⎢

⎣ 1 0 0

1 ⎤ �

⎦

Since X · Y = Y · X = I, X and Y are inverse matrices.

Identity Matrix for

Multiplication

If A is an n × n matrix and I is the identity matrix,

then A � I = A and I � A = A.

Example

ExercisesDetermine whether the matrices in each pair are inverses of each other.

1. ⎡ ⎢

⎣ 4 3 5

4 ⎤ �

⎦ and

⎡ ⎢

⎣ 4 -3

-5 4 ⎤ �

⎦ 2.

⎡ ⎢

⎣ 3 5 2

4 ⎤ �

⎦ and

⎡

⎢

⎣

2 - 5 −

2 -1

3 − 2

⎤

�

⎦

3. ⎡ ⎢

⎣ 2 5 3

-1 ⎤ �

⎦ and

⎡ ⎢

⎣ 2 -1

3 -2

⎤ �

⎦

4. ⎡ ⎢

⎣ 8 3 11

14 ⎤ �

⎦ and

⎡ ⎢

⎣ -4

3 11

-8 ⎤ �

⎦ 5.

⎡ ⎢

⎣ 4 5 -1

3 ⎤ �

⎦ and

⎡ ⎢

⎣ 1 3 2

8 ⎤ �

⎦

6. ⎡ ⎢

⎣ 5 11

2 4 ⎤ �

⎦ and

⎡

⎢

⎣ -2 11 − 2 1

- 5 − 2 ⎤

�

⎦

7. ⎡ ⎢

⎣ 4 6 2

-2 ⎤ �

⎦ and

⎡

⎢

⎣

- 1 −

5

3 − 10

- 1 − 10

1 − 10

⎤

�

⎦

8.

⎡ ⎢

⎣ 5 4 8

6 ⎤ �

⎦ and ⎡

⎢ ⎣

-3

2 4

- 5 − 2 ⎤ � ⎦

9.

⎡ ⎢

⎣ 3 2 7

4 ⎤ �

⎦ and

⎡

⎢

⎣ 7 − 2

1

- 3 − 2

-2 ⎤

�

⎦

10. ⎡ ⎢

⎣ 3 4 2

-6 ⎤ �

⎦ and

⎡ ⎢

⎣ 3 -4

2 -3

⎤ �

⎦ 11.

⎡ ⎢

⎣ 7 17

2 5 ⎤ �

⎦ and

⎡ ⎢

⎣ 5 -17

-2 7 ⎤ �

⎦

12. ⎡ ⎢

⎣ 4 7 3

5 ⎤ �

⎦ and

⎡ ⎢

⎣ -5

7 3

-4 ⎤ �

⎦

Study GuideInverse Matrices and Systems of Equations

4-6 SCS MBC.N.2.1, MBC.A.2.1


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NAME DATE PERIOD

1st Pass


Example

3x - 7y = 12x + 5y = -8

Determine the coefficient, variable, and constant matrices.

⎡ ⎢

⎣ 3 1 -7

5 ⎤ �

⎦ ·

⎡ ⎢

⎣ x y

⎤ �

⎦ =

⎡ ⎢

⎣ 12

-8 ⎤ �

⎦

Find the inverse of the coefficient matrix.

1 − 3(5) - 1(-7)

⎡ ⎢

⎣

5

-1 7

3 ⎤ �

⎦ =

⎡

⎢

⎣

5 − 22

- 1 − 22

7 − 22

3 − 22

⎤

�

⎦

Rewrite the equation in the form of X = A-1B

⎡ ⎢

⎣ x y

⎤ �

⎦ =

⎡

⎢

⎣

5 − 22

- 1 − 22

7 − 22

3 − 22

⎤

�

⎦

⎡

⎢

⎣ 12

-8 ⎤ �

⎦

Solve.

⎡ ⎢

⎣ x y

⎤ �

⎦ =

⎡

⎢

⎣

2 − 11

- 18 − 11

⎤

�

⎦

Matrix Equations A matrix equation for a system of equations consists of the product of the coefficient and variable matrices on the left and the constant matrix on the right of the equals sign.

Use a matrix equation to solve a system of equations.

ExercisesUse a matrix equation to solve each system of equations.

1. 2x + y = 8 2. 4x - 3y = 185x - 3y = -12 x + 2y = 12

3. 7x - 2y = 15 4. 4x - 6y = 203x + y = -10 3x + y + 8= 0

5. 5x + 2y = 18 6. 3x - y = 24x = -4y + 25 3y = 80 - 2x


Inverse Matrices and Systems of Equations



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