12 Suppose A, B, and C are 2 3 2 matrices, E, F, and G are 3 3 3 matrices, H and K are 2 3 3 matrices, and L and M are 3 3 2 matrices. For each of the following, if the operation is defined, specify the size of the matrix that results. 1. (a) A 1 4C (b) 1 2K 1 L (c) 5K 1 3H (d) 0G 1 3sE 1 Fd (e) 3A 1 6sB 1 Md (f) 12M 2 L (g) F 1 G 2 2C (h) F 1 G, where and are scalars 2. (a) AB 1 C (b) 3GF (c) CK 1 B (d) CK 1 H (e) EMC (f) GLH (g) HLG (h) 2EL 1 5MB
Read moreTable of Contents
1
Functions and Sequences
1.1
Four Ways to Represent a Function
1.2
A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Exponential Functions
1.5
Logarithms; Semilog and Log-Log Plots
1.6
Sequences and Difference Equations
2
Limits
2.1
Limits of Sequences
2.2
Limits of Functions at Infinity
2.3
Limits of Functions at Finite Numbers
2.4
Limits: Algebraic Methods
2.5
Continuity
3
Derivatives
3.1
Derivatives and Rates of Change
3.2
The Derivative as a Function
3.3
Basic Differentiation Formulas
3.4
The Chain Rule
3.5
The Chain Rule
3.6
Exponential Growth and Decay
3.7
Derivatives of the Logarithmic and Inverse Tangent Functions
3.8
Linear Approximations and Taylor Polynomials
4
Applications of Derivatives
4.1
Maximum and Minimum Values
4.2
How Derivatives Affect the Shape of a Graph
4.3
LHospitals Rule: Comparing Rates of Growth
4.4
Optimization Problems
4.5
Recursions: Equilibria and Stability
4.6
Antiderivatives
5
Integrals
5.1
Areas, Distances, and Pathogenesis
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
The Substitution Rule
5.5
Integration by Parts
5.6
Partial Fractions
5.7
Integration Using Tables and Computer Algebra Systems
5.8
Improper Integrals
6
Applications of Integrals
6.1
Areas Between Curves
6.2
Average Values
6.3
Further Applications to Biology
6.4
Volumes
7
Differential Equations
7.1
Modeling with Differential Equations
7.2
Phase Plots, Equilibria, and Stability
7.3
Direction Fields and Eulers Method
7.4
Separable Equations
7.5
Phase Plane Analysis
7.6
Phase Plane Analysis
8
Vectors and Matrix Models
8.1
Coordinate Systems
8.2
Vectors
8.3
The Dot Product
8.4
Matrix Algebra
8.5
Matrices and the Dynamics of Vectors
8.6
Eigenvectors and Eigenvalues
8.7
Eigenvectors and Eigenvalues
8.8
Iterated Matrix Models
9
Multivariable Calculus
9.1
Functions of Several Variables
9.2
Partial Derivatives
9.3
Tangent Planes and Linear Approximations
9.4
The Chain Rule
9.5
Directional Derivatives and the Gradient Vector
9.6
Maximum and Minimum Values
10
Systems of Linear Differential Equations
10.1
Qualitative Analysis of Linear Systems
10.2
Qualitative Analysis of Linear Systems
10.3
Applications
10.4
Systems of Nonlinear Differential Equations
Textbook Solutions for Biocalculus: Calculus for Life Sciences
Chapter 8.4 Problem 6
Question
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
Solution
The first step in solving 8.4 problem number 6 trying to solve the problem we have to refer to the textbook question: 610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
From the textbook chapter Matrix Algebra you will find a few key concepts needed to solve this.
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Title
Biocalculus: Calculus for Life Sciences 1
Author
James Stewart, Troy Day
ISBN
9781133109631
Answer: 610 The notation Ak means the matrix A multiplied with itself k times. 7. (a)
Chapter 8.4 textbook questions
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Chapter 8: Problem 1 Biocalculus: Calculus for Life Sciences 1
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Chapter 8: Problem 2 Biocalculus: Calculus for Life Sciences 1
12 Suppose A, B, and C are 2 3 2 matrices, E, F, and G are 3 3 3 matrices, H and K are 2 3 3 matrices, and L and M are 3 3 2 matrices. For each of the following, if the operation is defined, specify the size of the matrix that results. 1. (a) A 1 4C (b) 1 2K 1 L (c) 5K 1 3H (d) 0G 1 3sE 1 Fd (e) 3A 1 6sB 1 Md (f) 12M 2 L (g) F 1 G 2 2C (h) F 1 G, where and are scalars 2. (a) AB 1 C (b) 3GF (c) CK 1 B (d) CK 1 H (e) EMC (f) GLH (g) HLG (h) 2EL 1 5MB
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Chapter 8: Problem 3 Biocalculus: Calculus for Life Sciences 1
Using the given matrices, calculate the quantities, if they are defined. A c 2 5 1 7d B c 7 x a 5 d C c 9 2 7 10d E 0 3 1 7 6 0 9 13 5 F x 2 9 6 y 13 0 1 0 G 13 2 0 5 9 12 7 0 1 H c 3 1 7 15 0 2d K c 1 0 1 0 y 1 d (a) A 2 3C (b) 3F 1 G 2 E (c) 5K 1 9H (d) H 2 12G (e) 5B 2 A (f) B 2 9K (g) 3F 2 F (h) H 2 K 1 2H
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Chapter 8: Problem 4 Biocalculus: Calculus for Life Sciences 1
Using the given matrices, calculate the quantities, if they are defined. A c 7 0 1 9d B c 0 2 13 6d C c 15 0 0 2d E x 5 3 0 9 1 6 4 0 F y 0 y 0 1 0 0 1 1 G 1 2 3 5 4 6 9 7 8 H c 2 1 3 3 7 6d K 9 2 1 5 7 8 (a) A 2 2BC (b) EG (c) FK 2 2K (d) HEA (e) 3BC 2 HK (f) 7K 2 H (g) AHKB (h) BKHA 5. Find the transpose of each matrix. (a) 3X 1 2 (b) f3 3 9g (c) c 2 1 7 8 3 6d (d) c 2 1 1 3
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Chapter 8: Problem 5 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 6. (a) For the 2 3 2 identity matrix I, show that I 2 I. (b) For the n 3 n identity matrix I, show that I 2 I. (c) What do you think the entries of I k are?
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Chapter 8: Problem 6 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
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Chapter 8: Problem 7 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
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Chapter 8: Problem 8 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
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Chapter 8: Problem 9 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
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Chapter 8: Problem 10 Biocalculus: Calculus for Life Sciences 1
610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.
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Chapter 8: Problem 11 Biocalculus: Calculus for Life Sciences 1
For arbitrary 2 3 2 matrices A, B, and C, verify the following properties. (a) AsBCd sABdC (b) AsB 1 Cd AB 1 AC (c) sB 1 CdA BA 1 CA
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Chapter 8: Problem 12 Biocalculus: Calculus for Life Sciences 1
Suppose that A and B are arbitrary 2 3 2 matrices. Is the quantity given always equal to sA 1 Bd 2 ? (a) A2 1 2AB 1 B2 (b) sB 1 Ad 2 (c) AsA 1 Bd 1 BsA 1 Bd (d) A2 1 AB 1 BA 1 B
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Chapter 8: Problem 13 Biocalculus: Calculus for Life Sciences 1
Suppose that A is a 2 3 2 matrix that commutes with all possible 2 3 2 matrices. Show that, necessarily, a11 a22 and a12 a21 0. Hint: Try using X c 1 0 0 0d and Y c 0 1 0 0d as two particular 2 3 2 matrices with which A must commute.
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Chapter 8: Problem 14 Biocalculus: Calculus for Life Sciences 1
a) For any 2 3 2 matrices A and B show that sA 1 Bd T AT 1 BT (b) For any n 3 n matrices A and B show that sA 1 Bd T AT 1 BT
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Chapter 8: Problem 15 Biocalculus: Calculus for Life Sciences 1
(a) For any 2 3 2 matrices A and B show that sABd T BT AT (b) For any n 3 n matrices A and B show that sABd T BT AT
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Chapter 8: Problem 16 Biocalculus: Calculus for Life Sciences 1
A matrix A is called symmetric if AT A. Verify, for all 3 3 3 matrices B, that B 1 BT is symmetric
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Chapter 8: Problem 17 Biocalculus: Calculus for Life Sciences 1
Suppose a matrix A satisfies the equation A 1 AT 0. What must be true about the entries of A? Such matrices are called skew-symmetric
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Chapter 8: Problem 18 Biocalculus: Calculus for Life Sciences 1
An upper triangular matrix is a square matrix with all zeros below its diagonal. A lower triangular matrix is a square matrix with all zeros above its diagonal. (a) For all 3 3 3 upper triangular matrices U, verify that U 2 is also upper triangular. (b) For all 3 3 3 lower triangular matrices L, verify that L2 is also lower triangular.
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