(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) b) c) 14 1 / 5 2 3 4.7 3 1 0 0 5 4 2/3 9 0 12 13 0 1/2 11 14 2 0 0 0 0 1 2 3 0 1 4 0 0 1 3x y = 18 2) Consider the system. (A-D) x + 2y = 1 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.)
(Exercises for Chapter 8: Matrices and Determinants) E.8.2 In Exercises 3-12, use matrices and Gaussian Elimination with Back-Substitution. 4x + 2y = 3 3) Solve the system. (A-D) x + y = 2 3x 1 9x 2 = 57 4) Solve the system. (A-D) 5x 1 + 4x 2 = 18 5) ADDITIONAL PROBLEM: Solve the system x + 3y = 6 6) Solve the system. (A-E) 2x 6y = 9 2a + 3b = 16 3 1 2 a 4b = 17 3. (A-D) 7) Consider the system have? (A-E) x + 3y = 6. How many solutions does the system 2x 6y = 12 8) Solve the system 9) Solve the system y 2z = 14 x + z = 3. Begin by rewriting the system. (A-D) 4x + 6z = 22 3x 10y + 9z = 50 2x + 6y z = 27. (A-D) x 2y z = 10 10) Solve the system 11) Solve the system 12) Solve the system a 4b 3c = 5 a 4b c = 2. (A-D) 2a 7b 4c = 7 2x 1 + 8x 2 10x 3 = 20 3x 1 + 5x 2 + x 3 = 5. (A-D) 4x 1 x 2 + 3x 3 = 12 x 1 2x 2 x 3 = 3 5x 1 10x 2 5x 3 = 11. (A-E) 4x 1 + 3x 2 + 2x 3 = 18
(Exercises for Chapter 8: Matrices and Determinants) E.8.3 1 0 7 0 1 13) Consider the augmented matrix 0 1 3 0 2. (F, G) 0 0 0 1 4 a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? 1 4 1 0 2 0 0 1 0 14) Consider the augmented matrix 1. (F, G) 0 0 0 1 0 0 0 0 0 0 a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? 1 1 1 1 15) Consider the augmented matrix 0 2 4 3. (F, G) 0 0 3 6 a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form? 1 0 2 1 0 0 1 16) Consider the augmented matrix 2. (F, G) 0 0 1 3 0 0 0 0 a) Is the matrix in Row-Echelon Form? b) Is the matrix in Reduced Row-Echelon (RRE) Form?
(Exercises for Chapter 8: Matrices and Determinants) E.8.4 17) Solve the system 2x + 6y = 6 4x + 13y = 14 using Gauss-Jordan Elimination. (A-H) x z = 3 18) Solve the system 2y = 16 3x + z = 13 the system. (A-H) using Gauss-Jordan Elimination. Begin by rewriting 19) Solve the system 4x 1 + 4x 2 3x 3 = 7 5x 1 + 7x 2 13x 3 = 9 using Gauss-Jordan Elimination. (A-H) x 1 + 2x 2 5x 3 = 6
(Exercises for Chapter 8: Matrices and Determinants) E.8.5 SECTION 8.2: OPERATIONS WITH MATRICES Assume that all entries (i.e., elements) of all matrices discussed here are real numbers. 1 3 7 0 1) Let A = 2 4 and B = 8 3. (C) 7 9 5 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 6 4 1 4 3 and D = 2 3 1. Find the indicated matrix products. 2 1 0 If the matrix product is undefined, write Undefined. (D, E) a) CD b) DC c) D 2, 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 c 56.
(Exercises for Chapter 8: Matrices and Determinants) E.8.6 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) 2 0 0 7) Let D = 0 3 0. This is called a diagonal matrix. (D, E) 0 0 4 a) Find D 2. b) Based on a), conjecture (guess) what D 10 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 I 4. (F) 4x 1 + 4x 2 3x 3 = 7 10) In Section 8.1, Exercise 19, you solved the system 5x 1 + 7x 2 13x 3 = 9. x 1 + 2x 2 5x 3 = 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?
(Exercises for Chapter 8: Matrices and Determinants) E.8.7 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 AA1.) b) A 1, where A = 2 4 6 12. 0 4 8 c) A 1, where A = 0 0 3 5 0 0 2 13 5 d) A 1, where A = 1 5 2 1 2 8 3x 1 x 2 = 15 3) Use Exercise 2a to solve the system. (D) x 1 + 2x 2 = 2 4) Use Exercise 2d to solve the system 2x 1 +13x 2 5x 3 = 7 x 1 + 5x 2 + 2x 3 = 1. (D) x 1 2x 2 8x 3 = 7 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). (Exercises for Chapter 8: Matrices and Determinants) E.8.8 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 1 A 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)
(Exercises for Chapter 8: Matrices and Determinants) E.8.9 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 30 20 d) Let D = 5 4. Find det( C), or C. Compare with b).. 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 = B T, the transpose of B. Rows become columns, and vice-versa. f) Find g) Find h) Find 2 4 3 7. 4 5 0 0. 4 5 40 50. i) Find a ca b cb. Compare with g) and h). j) Find 1 ex x e 2 x. Here, the entries correspond to functions. k) Find sin cos cos. Here, the entries correspond to functions. sin
2 4 3 2) Let A = 1 3 3 5 1 2 (Exercises for Chapter 8: Matrices and Determinants) E.8.10. 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. 1 5 2 3) Let B = 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. 1 2 3 4) Find 10 20 30. Based on this exercise and Exercise 1i, make a conjecture (guess) 4 5 6 about determinants. (B, C) 10 400 500 5) Let C = 0 20 600. C is called an upper triangular matrix. Find det( C), or 0 0 30 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)
(Exercises for Chapter 8: Matrices and Determinants) E.8.11 7) Find 3 13 42 5 92 5 3.2 e e 2 267 9876 0 0 0 0. (C) 4 2 8) Solve the equation = 0 for (lambda). In doing so, you are finding the 1 1 4 2 eigenvalues of the matrix 1 1. SECTION 8.5: APPLICATIONS OF DETERMINANTS 4x + 2y = 3 1) Use Cramer s Rule to solve the system. x + y = 2 This was the system in Section 8.1, Exercise 3. (A) 3x 1 x 2 = 15 2) Use Cramer s Rule to solve the system. x 1 + 2x 2 = 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 Distances and lengths are measured in meters. (B) ( ) in the xy-plane.
(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 18 1 2 1 b) 2 2 (order 2 square) c) 2 1 d) e) f) g) 1 2 3 1 1 2 0 7 1 2 0 1 1 18 1 21 1 3 x + 2y = 1 y = 3 h) {( 5, 3) } i) 35 ( )( 3)= 18 ( 5)+ 2( 3)= 1
(Answers for Chapter 8: Matrices and Determinants) A.8.2 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) }
1) a) b) (Answers for Chapter 8: Matrices and Determinants) A.8.3 SECTION 8.2: OPERATIONS WITH MATRICES 6 3 10 1 2 + 5 31 9 26 24 57 3 4 5 2) [ 22], or simply the scalar 22 (depending on context) 3) 4) a) b) c) No a) 7 2 7 6 4 1 8 9 16 5 8 15 9 0 b) Undefined c) 3 20 10 6 4 5 4 9 7
(Answers for Chapter 8: Matrices and Determinants) A.8.4 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) b) 4 0 0 0 9 0 0 0 16 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
(Answers for Chapter 8: Matrices and Determinants) A.8.5 10) a) b) c) d) 4 4 3 A = 5 7 13 1 2 5 X = x 1 x 2 x 3 7 B = 9 6 The coefficient matrix was I 3. 1 The RHS was 5, which is the solution for X. 3
(Answers for Chapter 8: Matrices and Determinants) A.8.6 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX 1) I 2, which is 2) a) 2 7 1 7 1 7 3 7 1 0 0 1 b) A is noninvertible c) d) 0 0 1 5 1 2 0 4 3 1 0 0 3 4 38 3 2 7 3 3 1 1 3 17 3 1 1 3 3) {( 4, 3) }. Hint: Use X = A 1 B. 4) {( 1, 0, 1) }. Hint: Use X = A 1 B. 5) a) 7; b) 2 7 1 7 1 7, just as in Exercise 2a 3 7 6) a) 0; b) A 1 is undefined, just as in Exercise 2b 7) Hint: Show that ( AB) ( B 1 A 1 )= I n, or B 1 A 1 8) Hint: Show that AA 1 = I 2. ( ) AB ( )= I n.
(Answers for Chapter 8: Matrices and Determinants) A.8.7 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)= 10 det( B). e) 2. det( E)= det( B T )= 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) e 2 x xe x, or e x ( e x x) k) 1 2) a) 92. Hint: + ( 12)+ ( 60)+ ( 3)( 45) ( 6)( 8)= 92. b) 92. Hint: + ( 2) ( 9) ( 4) ( 17)+ 314 ( )= 92. c) 92. Hint: ( 4) ( 17)+ ( 3) ( 11) ()9 1 ( )= 92. 3) a) 17. Hint: + ( 12)+ ( 0)+ ( 24) ( 8) ( 0) ( 45)= 17. b) 17. Hint: 37 ( )+ ( 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.
(Answers for Chapter 8: Matrices and Determinants) A.8.8 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. 1) SECTION 8.5: APPLICATIONS OF DETERMINANTS 1 2, 5 2 2) {( 4, 3) } 3) a) 20 square meters (or m 2 ) b) 29 square meters (or m 2 ) c) 0. This implies that the vectors are parallel (i.e., the position vectors are collinear). 4) 12 square meters (or m 2 )