CHAPTER 8: Matrices and Determinants - Math Notes and Math ... · (Exercises for Chapter 8: Matrices and Determinants) E.8.3 13) Consider the augmented matrix 10 70 01 3 0 0001 1

(Exercises for Chapter 8: Matrices and Determinants) E.8.1

CHAPTER 8:

Matrices and Determinants

(A) means “refer to Part A,” (B) means “refer to Part B,” etc.

Most of these exercises can be done without a calculator, though one may be permitted.

SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS

1) Give the size and the number of entries (or elements) for each matrix below. (A)

a)

14 1 / 5

2 3

4.7

b)

3 1 0 0 5

4 2 / 3 9 0 12

13 0 1 / 2 11 14

2 0 0 0 0

c)

1 2 3

0 1 4

0 0 1

2) Consider the system 3x y = 18

x + 2y = 1. (A-D)

a) Write the augmented matrix for the given system.

b) What size is the coefficient matrix?

c) What size is the right-hand side (RHS)?

d) Switch Row 1 and Row 2 of the augmented matrix. Write the new matrix.

e) Take the matrix from d) and add 3( ) times Row 1 to Row 2. Write the new

matrix.

f) Take the matrix from e) and divide Row 2 by 7( ) ; that is, multiply Row 2 by

1

7. Write the new matrix, which will be in Row-Echelon Form (Part F).

g) Write the system corresponding to the matrix from f).

h) Solve the system from g) using Back-Substitution, and write the solution set.

i) Check your solution in the original system. (This is typically an optional step.)


In Exercises 3-12, use matrices and Gaussian Elimination with Back-Substitution.

3) Solve the system 4x + 2y = 3

x + y = 2. (A-D)

4) Solve the system 3x1 9x2 = 57

5x1 + 4x2 = 18. (A-D)

5) ADDITIONAL PROBLEM: Solve the system

2a + 3b =16

31

2a 4b =

17

3

. (A-D)

6) Solve the system x + 3y = 6

2x 6y = 9. (A-E)

7) Consider the system x + 3y = 6

2x 6y = 12. How many solutions does the system

have? (A-E)

8) Solve the system

y 2z = 14

x + z = 3

4x + 6z = 22

. Begin by rewriting the system. (A-D)

9) Solve the system

3x 10y + 9z = 50

2x + 6y z = 27

x 2y z = 10

. (A-D)

10) Solve the system

a 4b 3c = 5

a 4b c = 2

2a 7b 4c = 7

. (A-D)


2x1 + 8x2 10x3 = 20

3x1 + 5x2 + x3 = 5

4x1 x2 + 3x3 = 12

. (A-D)


x1 2x2 x3 = 3

5x1 10x2 5x3 = 11

4x1 + 3x2 +2x3 = 18

. (A-E)


13) Consider the augmented matrix

1 0 7 0

0 1 3 0

0 0 0 1

1

2

4

. (F, G)

a) Is the matrix in Row-Echelon Form?

b) Is the matrix in Reduced Row-Echelon (RRE) Form?


1 4 1 0

0 0 1 0

0 0 0 1

0 0 0 0

2

1

0

0

. (F, G)




1 1 1

0 2 4

0 0 3

1

3

6

. (F, G)




1 0 2

0 0 1

0 0 1

0 0 0

1

2

3

0

. (F, G)




17) Solve the system 2x + 6y = 6

4x +13y = 14 using Gauss-Jordan Elimination. (A-H)


x z = 3

2y = 16

3x + z = 13

using Gauss-Jordan Elimination. Begin by rewriting

the system. (A-H)


4x1 + 4x2 3x3 = 7

5x1 + 7x2 13x3 = 9

x1 + 2x2 5x3 = 6

using Gauss-Jordan Elimination. (A-H)


SECTION 8.2: OPERATIONS WITH MATRICES

Assume that all entries (i.e., elements) of all matrices discussed here are real numbers.

1) Let A =

1 3

2 4

7

and B =

7 0

8 3

9 5

. (C)

a) Find A + B .

b) Find 3A 4B .

2) Let A = 3 2[ ] and B =4

5. Find AB . (D)

3) Let A =4 1

2 3 and B =

1 0

3 2. (B, D, E)

a) Find AB.

b) Find BA.

c) Yes or No: Is AB = BA here?

4) Let C =2 0 7

1 4 3 and D =

1 6 4

2 3 1

2 1 0

. Find the indicated matrix products.

If the matrix product is undefined, write “Undefined.” (D, E)

a) CD

b) DC

c) D2, which is defined to be DD

5) Assume that A is an 8 10 matrix and B is a 10 7 matrix. (D, E)

a) What is the size of the matrix AB?

b) Let C = AB . Explain how to obtain the matrix element c56 .


6) If A is an m n matrix and B is a p q matrix, under what conditions are both AB

and BA defined? (E)

7) Let D =

2 0 0

0 3 0

0 0 4

. This is called a diagonal matrix. (D, E)

a) Find D2.

b) Based on a), conjecture (guess) what D10 is. (Calculator)

8) Assume that A is a 3 4 matrix, B is a 3 4 matrix, and C is a 4 7 matrix.

Find the sizes of the indicated matrices. If the matrix expression is undefined, write

“Undefined.” (C-E)

a) A + 4B

b) A C

c) AB

d) AC

e) AC + BC

f) A + B( )C

9) Write the identity matrix I4 . (F)

10) In Section 8.1, Exercise 19, you solved the system

4x1 + 4x2 3x3 = 7

5x1 + 7x2 13x3 = 9

x1 + 2x2 5x3 = 6

.

This system can be written in the form AX = B . (G)

a) Identify A.

b) Identify X (in the given system, before it is solved).

c) Identify B.

d) When you solved this system using Gauss-Jordan Elimination, what was the

coefficient matrix of the final augmented matrix in Reduced Row-Echelon

(RRE) Form? What was the right-hand side (RHS) of that matrix?


SECTION 8.3: THE INVERSE OF A SQUARE MATRIX

Assume that all entries (i.e., elements) of all matrices discussed here are real numbers.

1) If A is an invertible 2 2 matrix, what is AA 1? (B)

2) Find the indicated inverse matrices using Gauss-Jordan Elimination. If the inverse

matrix does not exist, write “A is noninvertible.” (C)

a) A 1, where A =

3 1

1 2. (Also check by finding AA 1

.)

b) A 1, where A =

2 4

6 12.

c) A 1, where A =

0 4 8

0 0 3

5 0 0

d) A 1, where A =

2 13 5

1 5 2

1 2 8

3) Use Exercise 2a to solve the system 3x1 x2 = 15

x1 + 2x2 = 2. (D)

4) Use Exercise 2d to solve the system

2x1 +13x2 5x3 = 7

x1 + 5x2 + 2x3 = 1

x1 2x2 8x3 = 7

. (D)

5) Let A =3 1

1 2, as in Exercise 2a. (E, F)

a) Find det A( ) .

b) Find A 1 using the shortcut from Part F. Compare with your answer to

Exercise 2a.


6) Let A =2 4

6 12, as in Exercise 2b. (E, F)

a) Find det A( ) .

b) What do we then know about A 1? Compare with your answer to Exercise 2b.

7) Assume that A and B are invertible n n matrices. Prove that AB( )1= B 1A 1 . (B)

8) Verify that the shortcut formula for A 1 given in Part F does, in fact, give the inverse

of a matrix A, where A =a b

c d and det A( ) 0 . (E, F)


SECTION 8.4: THE DETERMINANT OF A SQUARE

MATRIX

Assume that all entries (i.e., elements) of all matrices discussed here are real numbers,

unless otherwise indicated.

1) Find the indicated determinants. (B)

a) Let A = 4[ ] . Find det A( ) , or A .

b) Let B =3 2

5 4. Find det B( ) , or B .

c) Let C =5 4

3 2. Find det C( ) , or C . Compare with b).

d) Let D =30 20

5 4. Find det D( ) , or D . Compare with b).

e) Let E =3 5

2 4. Find det E( ) , or E . Compare with b).

Note: E = BT, the transpose of B. Rows become columns, and vice-versa.

f) Find 2 4

3 7.

g) Find 4 5

0 0.

h) Find 4 5

40 50.

i) Find a b

ca cb. Compare with g) and h).

j) Find 1 ex

x e2x. Here, the entries correspond to functions.

k) Find sin cos

cos sin. Here, the entries correspond to functions.


2) Let A =

2 4 3

1 3 3

5 1 2

. We will find det A( ) , or A , in three different ways. (B, C)

a) Use Sarrus’s Rule, the shortcut for finding the determinant of a 3 3 matrix.

b) Expand by cofactors along the first row.

c) Expand by cofactors along the second column.

3) Let B =

1 5 2

3 4 0

1 4 3

. We will find det B( ) , or B , in three different ways. (B, C)

a) Use Sarrus’s Rule, the shortcut for finding the determinant of a 3 3 matrix.

b) Expand by cofactors along the second row.

c) Expand by cofactors along the third column.

4) Find

1 2 3

10 20 30

4 5 6

. Based on this exercise and Exercise 1i, make a conjecture (guess)

about determinants. (B, C)

5) Let C =

10 400 500

0 20 600

0 0 30

. C is called an upper triangular matrix. Find det C( ) , or

C . Based on this exercise, make a conjecture (guess) about determinants of upper

triangular matrices such as C. What about lower triangular matrices? (B, C)

6) Find

1 4 0 2

3 2 0 1

1 3 0 4

0 2 2 1

. (B, C)


7) Find

13 42 53

92 5 3.2 e

e2 267 9876

0 0 0 0

. (C)

8) Solve the equation 4 2

1 1= 0 for (lambda). In doing so, you are finding the

eigenvalues of the matrix 4 2

1 1.

SECTION 8.5: APPLICATIONS OF DETERMINANTS

1) Use Cramer’s Rule to solve the system 4x + 2y = 3

x + y = 2.

This was the system in Section 8.1, Exercise 3. (A)

2) Use Cramer’s Rule to solve the system 3x1 x2 = 15

x1 + 2x2 = 2.

This was the system in Section 8.3, Exercise 3. (A)

3) Find the area of the parallelogram determined by each of the following pairs of

position vectors in the xy-plane. Distances and lengths are measured in meters. (B)

a) 4, 0 and 0, 5

b) 2, 5 and 7, 3

c) 1, 4 and 3, 12 . (What does the result imply about the vectors?)

4) Find the area of the triangle with vertices 2, 1( ) , 3, 1( ) , and 1, 5( ) in the xy-plane.

Distances and lengths are measured in meters. (B)

(Answers for Chapter 8: Matrices and Determinants) A.8.1

CHAPTER 8:

Matrices and Determinants

SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS

1) a) 3 2 , 6 entries; b) 4 5 , 20 entries; c) 3 3 (order 3 square), 9 entries

2)

a)

3 1

1 2

18

1

b) 2 2 (order 2 square)

c) 2 1

d)

1 2

3 1

1

18

e)

1 2

0 7

1

21

f)

1 2

0 1

1

3

g)

x + 2y = 1

y = 3

h) 5, 3( ){ }

i)

3 5( ) 3( ) = 18

5( ) + 2 3( ) = 1


3) 1

2,5

2

4) 2, 7( ){ }

5) 2

3,4

3

6)

7) The system has infinitely many solutions.

Note 1: The solutions correspond to the points on the line x + 3y = 6 in the xy-plane.

Note 2: The solution set can be given by: x, y( ) 2 x + 3y = 6{ } .

8) 2, 4, 5( ){ }

9) 7, 2, 1( ){ }

10) 1

2, 0,

3

2

11) 1, 1, 3( ){ }

12)

13) a) Yes; b) Yes

14) a) Yes; b) No

15) a) No; b) No

16) a) No; b) No

17) 3, 2( ){ }

18) 4, 8, 1( ){ }

19) 1, 5, 3( ){ }


SECTION 8.2: OPERATIONS WITH MATRICES

1)

a)

6 3

10 1

2 + 5

b)

31 9

26 24

57 3 4 5

2) 22[ ] , or simply the scalar 22 (depending on context)

3)

a)

7 2

7 6

b)

4 1

8 9

c) No

4)

a)

16 5 8

15 9 0

b) Undefined

c)

3 20 10

6 4 5

4 9 7


5)

a) 8 7

b) Multiply the fifth row of A and the sixth column of B, in that order.

6) n = p and m = q

7)

a)

4 0 0

0 9 0

0 0 16

b)

1024 0 0

0 59,049 0

0 0 1,048,576

8)

a) 3 4

b) Undefined

c) Undefined

d) 3 7

e) 3 7

f) 3 7 ; in fact, the expressions in e) and f) are equivalent.

9)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1


10)

a)

A =

4 4 3

5 7 13

1 2 5

b)

X =

x1x2x3

c)

B =

7

9

6

d)

The coefficient matrix was I3 .

The RHS was

1

5

3

, which is the solution for X.


SECTION 8.3: THE INVERSE OF A SQUARE MATRIX

1) I2 , which is 1 0

0 1

2)

a)

2

7

1

71

7

3

7

b) A is noninvertible

c)

0 01

51

4

2

30

01

30

d)

438

3

17

32

3

7

31

1

31

1

3

3) 4, 3( ){ } . Hint: Use X = A1B .

4) 1, 0, 1( ){ } . Hint: Use X = A1B .

5) a) 7; b)

2

7

1

71

7

3

7

, just as in Exercise 2a

6) a) 0; b) A 1 is undefined, just as in Exercise 2b

7) Hint: Show that AB( ) B 1A 1( ) = In , or B 1A 1( ) AB( ) = In .

8) Hint: Show that AA 1= I2 .


SECTION 8.4: THE DETERMINANT OF A SQUARE

MATRIX

1)

a) 4

b) 2

c) 2 . C is obtained by switching the two rows of B, and det C( ) = det B( ) .

d) 20. D is obtained by multiplying the first row of B by 10, and det C( ) = 10det B( ) .

e) 2. det E( ) = det BT( ) = det B( ) .

f) 26

g) 0

h) 0

i) 0. (This is exemplified by g) and h), in which the second row is a multiple of the

first row. For both, a = 4 and b = 5 . In g), c = 0 . In h), c = 10 .)

j) e2x xex , or ex ex x( )

k) 1

2)

a) 92. Hint: + 12( ) + 60( ) + 3( ) 45( ) 6( ) 8( ) = 92 .

b) 92. Hint: + 2( ) 9( ) 4( ) 17( ) + 3 14( ) = 92 .

c) 92. Hint: 4( ) 17( ) + 3( ) 11( ) 1( ) 9( ) = 92 .

3)

a) 17 . Hint: + 12( ) + 0( ) + 24( ) 8( ) 0( ) 45( ) = 17 .

b) 17 . Hint: 3 7( ) + 4( ) 1( ) + 0 = 17 .

c) 17 . Hint: + 2( ) 8( ) 0 + 3( ) 11( ) = 17 .

4) 0. If one row of a square matrix is a multiple of another row of that matrix, then the

determinant of the matrix is 0.


5) 6000. The determinant of an upper (or lower) triangular matrix is equal to the product

of the entries along the main diagonal.

6) 66 . Hint: Expand by cofactors along the third column. 2( ) 33( ) = 66 .

7) 0. Hint: Expand by cofactors along the fourth row.

8) 2, 3{ } . That is, the eigenvalues are 2 and 3.

SECTION 8.5: APPLICATIONS OF DETERMINANTS

1) 1

2,5

2

2) 4, 3( ){ }

3)

a) 20 square meters (or m2)

b) 29 square meters (or m2)

c) 0. This implies that the vectors are parallel (i.e., the position vectors are collinear).

4) 12 square meters (or m2)

Documents

CHAPTER 8: Matrices and Determinants - Math Notes and Math ... · (Exercises for Chapter 8: Matrices and Determinants) E.8.3 13) Consider the augmented matrix 10 70 01 3 0 0001 1